On Sets Defining Few Ordinary Circles
On Sets Defining Few Ordinary Circles
Aaron Lin 0 1 2 3
Mehdi Makhul 0 1 2 3
Hossein Nassajian Mojarrad 0 1 2 3
Josef Schicho 0 1 2 3
Konrad Swanepoel 0 1 2 3
Frank de Zeeuw 0 1 2 3
0 Department of Mathematics, London School of Economics and Political Science , London WC2A 2AE , United Kingdom
1 Editor in Charge: János Pach
2 Department of Mathematics, EPFL , 1015 Lausanne , Switzerland
3 Research Institute for Symbolic Computation, Johannes Kepler University , 4040 Linz , Austria
An ordinary circle of a set P of n points in the plane is defined as a circle that contains exactly three points of P . We show that if P is not contained in a line or a circle, then P spans at least n2/4 − O (n) ordinary circles. Moreover, we determine the exact minimum number of ordinary circles for all sufficiently large n and describe all point sets that come close to this minimum. We also consider the circle variant of the orchard problem. We prove that P spans at most n3/24 − O (n2) circles passing through exactly four points of P . Here we determine the exact maximum and the extremal configurations for all sufficiently large n. These results are based on the

following structure theorem. If n is sufficiently large depending on K , and P is a set
of n points spanning at most K n2 ordinary circles, then all but O(K ) points of P lie
on an algebraic curve of degree at most four. Our proofs rely on a recent result of
Green and Tao on ordinary lines, combined with circular inversion and some classical
results regarding algebraic curves.
Mathematics Subject Classification 52C35 (52C10 14N10 14H50)
1 Introduction
1.1 Background
The classical Sylvester–Gallai theorem states that any finite noncollinear point set in
R2 spans at least one ordinary line (a line containing exactly two of the points). A
more sophisticated statement is the socalled Dirac–Motzkin conjecture, according to
which every noncollinear set of n > 13 points in R2 determines at least n/2 ordinary
lines. This conjecture was proved by Green and Tao [13] for all sufficiently large n.
Their proof was based on a structure theorem, which roughly states that any point set
with a linear number of ordinary lines must lie mostly on a cubic curve (see Theorem
5.1 for a precise statement).
It is natural to ask the corresponding question for ordinary circles (circles that
contain exactly three of the given points); see for instance [8, Sect. 7.2] or [17, Chap. 6].
Elliott [12] introduced this question in 1967, and proved that any n points, not all on
a line or a circle, determine at least 2n2/63 − O(n) ordinary circles. (Throughout
the paper, by O( f (n)) we mean a function g(n) such that 0 g(n) C f (n) for
some constant C > 0 and all sufficiently large n. Thus, −O(n) is a function g(n)
satisfying −C n g(n) 0 for sufficiently large n.) He suggested, cautiously, that
the optimal bound is n2/6 − O(n). Elliott’s result was improved by Bálintová and
Bálint [1, Rem., p. 288] to 11n2/247 − O(n), and Zhang [26] obtained n2/18 − O(n).
Zhang also gave constructions of point sets on two concentric circles with n2/4− O(n)
ordinary circles.
We will use the results of Green and Tao to prove that n2/4 − O(n) is
asymptotically the right answer, thus disproving the bound suggested by Elliott [12]. Nassajian
Mojarrad and de Zeeuw proved this bound in an earlier preprint [19], which is
subsumed by this paper, and will not be published independently. We will find the exact
minimum number of ordinary circles, for sufficiently large n, and we will determine
which configurations attain or come close to that minimum. We make no attempt to
specify the threshold implicit in the phrase ‘for sufficiently large n’; any improvement
would depend on an improvement of the threshold in the result of Green and Tao [13].
For small n, the bound 19 n2 due to Zhang [26] remains the best known lower bound
on the number of ordinary circles.
Green and Tao [13] also solved (for large n) the even older orchard problem, which
asks for the exact maximum number of lines passing through exactly three points of a
set of n points in the plane. We refer to [13] for the history of this problem. The upper
bound 13 n2 is easily proved by double counting, but it is not the exact maximum.
Using group laws on certain cubic curves, one can construct n noncollinear points
with n(n − 3)/6 + 1 3point lines, and Green and Tao [13] proved (for large n) that
this is optimal. This does not follow directly from the Dirac–Motzkin conjecture, but
it does follow from the abovementioned structure theorem of Green and Tao for sets
with few ordinary lines (Theorem 5.1).
The analogous orchard problem for circles asks for the maximum number of circles
passing through exactly four points from a set of n points. As far as we know, this
question has not been asked before. We determine the exact maximum and the extremal
sets for all sufficiently large n.
Although we do not consider other related problems, we remark that similar
questions have been asked for ordinary conics [7,10,25], ordinary planes [2], and ordinary
hyperplanes [3].
1.2 Results
Our first main result concerns the minimum number of ordinary circles spanned by a
set of n points, not all lying on a line or a circle, and the structure of sets of points
that come close to the minimum. The first part of the theorem solves Problem 6 in
[8, Sect. 7.2].
Theorem 1.1 (Ordinary circles)
(i) If n is sufficiently large, the minimum number of ordinary circles determined by
n points in R2, not all on a line or a circle, equals
if n ≡ 0 (mod 4),
if n ≡ 1 (mod 4),
if n ≡ 2 (mod 4),
if n ≡ 3 (mod 4).
(ii) Let C be sufficiently large. If a set P of n points in R2 determines fewer than
n2/2 − C n ordinary circles, then P lies on the disjoint union of two circles, or
the disjoint union of a line and a circle.
In Sect. 4, we will describe constructions that meet the lower bound in part (i) of
Theorem 1.1. For even n, the bound in part (i) is attained by certain constructions on the
disjoint union of two circles, while for odd n, the bound is attained by constructions
on the disjoint union of a line and a circle. The main tools in our proof are circle
inversion and the structure theorem of Green and Tao [13] for sets with few ordinary
lines, together with some classical results about algebraic curves and their interaction
with inversion.
Let us define a generalised circle to be either a circle or a line. Because inversion
maps circles and lines to circles and lines, it turns out that in our proof it is more natural
to work with generalised circles. Alternatively, we could phrase our results in terms
of the inversive plane (or Riemann sphere) R2 ∪ {∞}, where ∞ is a single point that
lies on all lines, which can then also be considered as circles. Yet another equivalent
view would be to identify the inversive plane with the sphere S2 via stereographic
projection, and consider circles on S2, which are in bijection with generalised circles.
All our statements about generalised circles in R2 could thus be formulated in terms
of circles in R2 ∪ {∞} or on S2.
We define an ordinary generalised circle to be one that contains three points from a
given set. Our proof of Theorem 1.1 proceeds via an analogous theorem for ordinary
generalised circles, which turns out to be somewhat easier to obtain.
Theorem 1.2 (Ordinary generalised circles)
(i) If n is sufficiently large, the minimum number of ordinary generalised circles
determined by n points in R2, not all on a generalised circle, equals
if n ≡ 0 (mod 4),
if n ≡ 1 (mod 4),
if n ≡ 2 (mod 4),
if n ≡ 3 (mod 4).
(ii) Let C be sufficiently large. If a set P of n points in R2 determines fewer than
n2/2 − C n ordinary generalised circles, then P lies on two disjoint generalised
circles.
We also solve the analogue of the orchard problem for circles (for sufficiently
large n). We define a 4point (generalised) circle to be a (generalised) circle that
passes through exactly four points of a given set of n points. The ‘circular cubics’ in
part (ii) will be defined in Sect. 2.
Theorem 1.3 (4Point generalised circles)
(i) If n is sufficiently large, the maximum number of 4point generalised circles
determined by a set of n points in R2 is equal to
⎧ 1
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨ 22144
1
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ 22144
if n ≡ 0
if n ≡ 1, 3, 5, 7
if n ≡ 2, 6
if n ≡ 4
(ii) Let C be sufficiently large. If a set P of n points in R2 determines more than
n3/24 − 7n2/24 + C n 4point generalised circles, then up to inversions, P lies
on an ellipse or a smooth circular cubic.
Theorem 1.3 remains true if we replace ‘generalised circles’ by ‘circles’. This is
because we can apply an inversion to any set of n points with a maximum number
of generalised circles in such a way that all straightline generalised circles become
circles. Therefore, the maximum is also attained by circles only.
The proofs of the above theorems are based on the following structure theorems in
the style of Green and Tao [13]. The first gives a rough picture, by stating that a point
set with relatively few ordinary generalised circles must lie on a bicircular quartic, a
specific type of algebraic curve of degree four that we introduce in Sect. 2.
Theorem 1.4 (Weak structure theorem) Let K > 0 and let n be sufficiently large
depending on K . If a set P of n points in R2 spans at most K n2 ordinary generalised
circles, then all but at most O (K ) points of P lie on a bicircular quartic.
Ball [2] concurrently obtained a similar result as a consequence of a structure
theorem for ordinary planes in R3. He shows that n points with O (n2+1/6) ordinary
circles must lie mostly on a quartic curve.
We define bicircular quartics in Sect. 2; they can be reducible, so in Theorem 1.4
the set P may also lie mostly on a lowerdegree curve contained in a bicircular quartic.
Our proof actually gives a more precise list of possibilities. The curve that P mostly
lies on can be: a line; a circle; an ellipse; a line and a disjoint circle; two disjoint circles;
a circular cubic that is acnodal or smooth; or a bicircular quartic that is an inverse of
an acnodal or smooth circular cubic.
A more precise characterisation of the possible configurations with few ordinary
generalised circles is given in the following theorem. The group structures referred to
in the theorem are defined in Sect. 3; the circular points at infinity (α and β) referred to
in Case (iii) are introduced in Sect. 2; and the ‘aligned’ and ‘offset’ double polygons
are defined in Sect. 4.
Theorem 1.5 (Strong structure theorem) Let K > 0 and let n be sufficiently large
depending on K . If a set P of n points in R2 spans at most K n2 ordinary generalised
circles, then up to inversions and similarities, P differs in at most O (K ) points from
a configuration of one of the following types:
(i) a subset of a line;
(ii) a subgroup of an ellipse;
(iii) a coset H ⊕ x of a subgroup H of a smooth circular cubic, for some x such that
4x ∈ H ⊕ α ⊕ β, where α and β are the two circular points at infinity;
(iv) a double polygon that is ‘aligned’ or ‘offset’.
In Sect. 2, we carefully introduce circular cubics and bicircular quartics, and show
their connection to inversion. In Sect. 3, we define group laws on these curves, which
help us construct point sets with few ordinary (generalised) circles in Sect. 4. In Sect. 5,
which forms the core of our proof, we derive Theorems 1.4 and 1.5 from the structure
theorem of Green and Tao [13]. In Sect. 6, we combine the structure theorems with
our analysis of the constructions from Sect. 4 to establish the precise statements in
Theorems 1.1, 1.2, and 1.3.
2 Circular Curves and Inversion
The key tool in our proof is circle inversion, as it was in the earlier papers [1, 12, 26]
on the ordinary circles problem; the first to use circle inversion in Sylvester–Gallai
problems was Motzkin [18]. The simple reason for the relevance of circle inversion
is that if we invert in a point of the given set, an ordinary circle through that point is
turned into an ordinary line. Thus we can use results on ordinary lines, like those of
Green and Tao [13], to deduce results about ordinary circles. To do this successfully,
we need a thorough understanding of the effect of inversion on algebraic curves, and
in particular we need to introduce the special class of circular curves.
2.1 Circular Curves and Circular Degree
In this subsection, we work in the real projective plane RP2, and partly in the complex
projective plane CP2. See for instance [22, App. A] for an appropriate introduction
to projective geometry. We use the homogeneous coordinates [x : y : z] for points in
RP2 or CP2, and we think of the line with equation z = 0 as the line at infinity. An
affine algebraic curve in R2, defined by a polynomial f ∈ R[x , y], can be naturally
extended to a projective algebraic curve, by taking the zero set of the homogenisation
of f . This curve in RP2 then extends to CP2, by taking the complex zero set of the
homogenised polynomial.
We define the circular points to be the points
on the line at infinity in CP2. The circular points play a key role in this paper, due
to the fact that every circle contains both circular points. Moreover, any real conic
containing α and β is either a circle, or a union of a line with the line at infinity. We
could thus consider a generalised circle to be a conic that contains both circular points.
Definition 2.1 An algebraic curve in RP2 is circular if it contains α and β. For k 2,
an algebraic curve in RP2 is kcircular if it has singularities of multiplicity at least k
at both α and β.
A classical reference for circular curves is Johnson [16], while a more modern one
is Werner [24]. Let us make the definition more explicit in three concrete cases.
A generalised circle is an algebraic curve of degree two that contains α and β;
equivalently, it is a curve in RP2 defined by a homogeneous polynomial of the form
where t ∈ R, and ∈ R[x , y, z] is a nontrivial linear form. If t = 0, then the curve
is a circle, while if t = 0, the curve is the union of a line with the line at infinity.
A circular cubic is an algebraic curve of degree three that contains α and β;
equivalently, it is any curve in RP2 defined by a homogeneous polynomial of the form
where u, v ∈ R, and q ∈ R[x , y, z] is a nontrivial quadratic homogeneous
polynomial. Note that we do not require a circular cubic to be irreducible or smooth. For
instance, the union of a line and a circle is a circular cubic, and so is the union of any
conic with the line at infinity (take u = v = 0 in (1)).
A bicircular quartic is an algebraic curve of degree four that is 2circular;
equivalently, it is any curve in RP2 defined by a homogeneous polynomial of the form
t (x 2 + y2)2 + (u x + vy)(x 2 + y2)z + q(x , y, z)z2,
where t , u, v ∈ R, and q ∈ R[x , y, z] is a nontrivial homogeneous quadratic
polynomial (see [24, Sect. 8.2] for a proof that a quartic is 2circular if and only if its
equation has the form (2)). A noteworthy example of a bicircular quartic is a union
of two circles, for which it is easy to see that the curve has double points at α and β,
since both circles contain those points.
Every circular cubic is contained in a bicircular quartic, since for t = 0 in (2) we
get a union of a circular cubic and the line at infinity. A noncircular conic is also
contained in a bicircular quartic, since for t = u = v = 0 in (2) we get a union of a
conic and z2 = 0, which is a double line at infinity.
Definition 2.2 The circular degree of an algebraic curve γ in RP2 is the smallest k
such that γ is contained in a kcircular curve of degree 2k.
The circular degree is welldefined, since given any curve γ of degree k, we can
add k copies of the line at infinity, to get a kcircular curve of degree 2k.
For example, a line has circular degree one, since its union with the line at infinity
is a 1circular curve of degree two. A conic that is not a circle has circular degree two,
since its union with two copies of the line at infinity is a 2circular curve of degree
four. Similarly, a circular cubic has circular degree two, since its union with the line at
infinity is a 2circular curve of degree four. We can thus classify curves of low circular
degree as follows:
• Circular degree one: lines and circles (that is, generalised circles).
• Circular degree two: noncircular conics, circular cubics, and bicircular quartics.
• Circular degree three: noncircular cubics, circular quartics, 2circular quintics,
and 3circular sextics.
This classification is important to us, because we will see that circular degree is
invariant under inversion.
We have defined circular curves and circular degrees in the projective plane, because
that is their most natural setting. In the rest of the paper, to avoid confusion between
the projective and inversive planes, we will use these notions for curves in R2, with
the understanding that to inspect the definitions we should consider RP2 and CP2.
2.2 Inversion
Circular curves are intimately related to circle inversion, which we now introduce. A
general reference is [6].
Definition 2.3 Let C ( p, r ) be the circle with centre p = (x p, yp) ∈ R2 and radius
r > 0. The circle inversion with respect to C ( p, r ) is the mapping I p,r : R2\{ p} →
R2\{ p} defined by
I p,r (x , y) =
r 2(x − x p)
r 2(y − yp)
for (x , y) = p. We write I p for I p,1. We call p the centre of the inversion I p,r .
In the inversive plane R2 ∪ {∞}, the inversion map can be completed by setting
I p,r ( p) = ∞ and I p,r (∞) = p, so that inversions take generalised circles to
generalised circles. The group of transformations of the inversive plane generated by the
inversions and the similarities is called the inversive group. It is known that a bijection
of the inversive plane that takes generalised circles to generalised circles has to be an
element of this group, and that any element of this group is either a similarity or an
inversion followed by an isometry [9, Thm. 6.71].
The image of an algebraic curve in R2 under an inversion is also an algebraic curve,
in the following sense.
We refer to γ as the inverse of γ with respect to the circle C ( p, r ), and abuse notation
slightly by writing γ = I p,r (γ ). Also, since for different choices of radius r , I p,r (γ )
differs only by a dilatation in p, we will often only consider the inverse I p(γ ) =
I p,1(γ ) and refer to it as the inverse of γ in the point p.
If a curve has degree d, then its inverse has degree at most 2d [24, Thm. 4.14].
If γ is irreducible, then its inverse is also irreducible. Note that inverses of algebraic
curves can behave somewhat unintuitively; for instance, Proposition 2.6 states that
the inverse of an ellipse has an isolated point, which is surprising if one thinks of an
ellipse as just a closed continuous curve.
It is well known that the inverses of generalised circles are again generalised circles.
It turns out that, more generally, circular degree is preserved under inversion. We now
make precise what this means for curves of low circular degree. A proof can be found
in the classical paper [16]; for a more modern reference, see [24, Sect. 9.2].
(i) The inverse of C1 in a point on C1 is a line; the inverse of C1 in a point not on
C1 is a circle.
(ii) The inverse of C2 in a singular point on C2 is a noncircular conic; the inverse
of C2 in a regular point on C2 is a circular cubic; the inverse of C2 in a point
not on C2 is a bicircular quartic.
(iii) The inverse of C3 in a singularity of multiplicity three is a noncircular cubic;
the inverse of C3 in a singularity of multiplicity two is a circular quartic; the
inverse of C3 in a regular point on C3 is a 2circular quintic; the inverse of C3
in a point not on C3 is a 3circular sextic.
One particular subcase of Case (ii) will play an important role in our paper, and
we state it separately in Proposition 2.6. A proof can be found in [14, p. 202]. Let us
recall that an acnodal cubic is a singular cubic with a singularity that is an isolated
point; for example, (2x − 1)(x 2 + y2) − y2 = 0 is an acnodal circular cubic with a
singularity at the origin.
Proposition 2.6 The inverse of an ellipse in a point on the ellipse is an acnodal
circular cubic with the centre of inversion as its singularity; the inverse of an acnodal
circular cubic in its singularity is an ellipse through the singularity.
For example, the inverse of the cubic (2x − 1)(x 2 + y2) − y2 = 0 in its singularity
at the origin is the ellipse (x − 1)2 + 2y2 = 1.
3 Groups on Circular Curves
3.1 Groups on Irreducible Circular Cubics
The extremal configurations in our main theorems are all based on group laws on
certain circular curves. It is well known that irreducible smooth cubics (elliptic curves)
have a group law (see for instance [22]). These groups play a crucial role in the work
of Green and Tao [13]. The reason that these groups are relevant to ordinary lines is
the following collinearity property of this group (when defined in the standard way).
Three points on the curve are collinear if and only if in the group they sum to the
identity element. For this property to hold, the identity element must be an inflection
point. Here we will define a group in a slightly different way (described for instance
in [22, Sect. 1.2]), in which the identity element is not necessarily an inflection point,
and the same collinearity property does not hold. However, for circular cubics, we
a ⊕ b
a ∗ b
show that we can choose the identity element so that we get a similar property for
concyclicity.
First let γ be any irreducible cubic, write γ ∗ for its set of regular points, and pick
an arbitrary point o ∈ γ ∗. We describe an additive group operation ⊕ on the set γ ∗ for
which o is the identity element. The construction is depicted in Fig. 1. Given a, b ∈ γ ∗,
let a ∗ b be the third intersection point of γ and the line ab, and define a ⊕ b to be
(a ∗ b) ∗ o, the third intersection point of γ and the line through a ∗ b and o. When
a = b, the line ab should be interpreted as the tangent line at a; when a ∗ b = o, the
line through a ∗ b and o should be interpreted as the tangent line to γ at o. We refer
to [22] for a more careful definition and a proof that this operation really does give a
group.
Now consider a circular cubic γ . Since the circular points α and β lying on it
are conjugate, γ has a unique real point on the line at infinity, which we choose as
our identity element o. We define the point ω to be the third intersection point of the
tangent line to γ at o (if there is no third intersection point, then o is an inflection point,
and we consider o itself to be the third point). Throughout this paper we will use ω to
denote this special point on a circular cubic; note that ω is not fixed like α and β, but
depends on γ . Also note that ω is real, since it corresponds to the third root of a real
cubic polynomial whose other two roots correspond to the real point o. Observe that
since α ∗ β = o, and by definition o ∗ o = ω.
With this group law, we no longer have the property that three points are collinear
if and only if they sum to o (unless o happens to be an inflection point). Nevertheless,
one can check that three points a, b, c ∈ γ ∗ are collinear if and only if a ⊕ b ⊕ c = ω.
d = d
a ∗ b
a ⊕ b
Fig. 2 Concyclicity of four regular points on a circular cubic
More important for us, four points of γ ∗ lie on a generalised circle if and only if
they sum to ω. This amounts to a classical fact (see [4, Art. 225] for an equivalent
statement), but we include a proof for completeness. We use the following version of
the Cayley–Bacharach Theorem, due to Chasles (see [11]).
Theorem 3.1 (Chasles) Suppose two cubic curves in CP2 with no common component
intersect in nine points, counting multiplicities. If γ is another cubic curve containing
eight of these intersection points, counting multiplicities, then γ also contains the
ninth.
Recall from Sect. 2 that a generalised circle, viewed projectively, is either a circle,
or the union a line with the line at infinity.
Proposition 3.2 Let γ be an irreducible circular cubic in RP2, and let a, b, c, d ∈ γ ∗
be points that are not necessarily distinct. A generalised circle intersects γ in the points
a, b, c, d (taking into account multiplicity) if and only if a ⊕ b ⊕ c ⊕ d = ω.
Proof We consider the cubic γ extended to CP2. We first show the forward direction.
All statements in the proof should be considered with multiplicity.
If the generalised circle is the union of a line and the line at infinity ∞, then
∪ ∞ intersects γ in a, b, c, d, α, β. Since intersects γ in at most three points,
one of the points a, b, c, d must equal o, say d = o. Since ∞ also intersects γ in at
most three points, we must have a, b, c ∈ . Thus a, b, c are collinear, and we have
a ⊕ b ⊕ c = ω, by the definition of the group law. It then follows from d = o that
a ⊕ b ⊕ c ⊕ d = ω.
Suppose next that the generalised circle is a circle σ , and intersects γ in
a, b, c, d, α, β. The construction that follows is depicted in Fig. 2. Let 1 be the line
through o and a ∗ b (and thus through a ⊕ b), 2 the line through a and b (and thus
through a ∗ b), and 3 the line through c and a ⊕ b. Note that σ and ∞ intersect in α
and β. Then γ1 = σ ∪ 1 and γ2 = 2 ∪ 3 ∪ ∞ are two cubic curves that intersect in
nine points, of which the eight points a, b, c, a ∗ b, a ⊕ b, o, α, and β certainly lie on
γ ; the remaining point is the third intersection point of γ1 and 3 beside c and a ⊕ b,
which we denote by d . By Theorem 3.1, γ contains d . By the group law on γ , we
have d = (a ⊕ b) ∗ c. Moreover, d must be the sixth intersection point of γ and σ
beside a, b, c, α, β, which is d, so d = d = (a ⊕ b) ∗ c. By the definition of the group
law, this implies a ⊕ b ⊕ c = o ∗ d, so (a ⊕ b ⊕ c) ∗ d = (o ∗ d) ∗ d = o, and finally
a ⊕ b ⊕ c ⊕ d = o ∗ o = ω.
For the converse, suppose that a ⊕ b ⊕ c ⊕ d = ω, and let d be the fourth point
where the generalised circle σ through a, b, c intersects γ . Then, by what we have
just shown, a ⊕ b ⊕ c ⊕ d = ω, and it follows that d = d , and a, b, c, d lie on σ .
This proposition is a consequence of the more general fact that six points on a
circular cubic lie on a conic if and only if they sum to 2ω. (In the standard group
structure on a cubic, where the identity o is chosen as an inflection point, they would
sum to o; see [23, Thm. 9.2].) Since a generalised circle in RP2 is a conic containing
α and β, and α ⊕ β = ω, it follows that four points a, b, c, d (possibly including o)
lie on a generalised circle if and only if they sum to ω.
We now define group laws on two other types of curves of circular degree two, and
observe that they satisfy similar concyclicity properties. Let us note at this point that
most bicircular quartics can also be given a group structure (if an irreducible bicircular
quartic has no singularities besides α and β, then it is a curve of genus one, and thus
has a group law by [21, Sect. III.3]). However, in our proofs we will handle bicircular
quartics by inverting in a point on the curve, which by Lemma 2.5 transforms a
bicircular quartic into a circular cubic. For that reason, we do not need to study the
group law on bicircular quartics separately.
Ellipses We discuss a group law on ellipses, although we do not actually need it
in our proof, because inversion lets us transform an ellipse into an acnodal cubic
(Proposition 2.6), which we have already given a group structure in the previous
subsection. Nevertheless, we treat the group law on ellipses here because it is especially
elementary, and it would be strange not to mention it.
Consider the ellipse σ given by the equation x 2 + (y/s)2 = 1, with s = 0, 1. For
any point a ∈ σ , we project a vertically to the point a on the unit circle around the
origin, as in Fig. 3, and call the angle θa the eccentric angle of a. We define the sum of
two points a, b ∈ σ to be the point c = a ⊕ b whose eccentric angle is θc = θa + θb.
This gives σ a group structure isomorphic to R/Z. The identity element is o = (1, 0),
and the inverse of a point is its reflection in the x axis. We have the following classical
fact that describes when four points on an ellipse are concyclic (see [15] for the oldest
reference we could find, and [5, Problem 17.2] for two detailed proofs).
Proposition 3.3 Four points a, b, c, d ∈ σ are concyclic if and only if a ⊕ b ⊕
c ⊕ d = o. We may allow two of the points to be equal, in which case the circle
through the three distinct points is tangent to the ellipse at the repeated point.
Another way to look at this group law is that we are parametrising the ellipse using
lines through o = (1, 0) (see for instance [22, Sect. 1.1]). More precisely, each point
a ∈ σ corresponds to the line oa; oa makes an angle π − θa /2 with the x axis, and the
set of lines through o thus has a group structure equivalent to the one above. This view
lets us relate the group on the ellipse to the group on the acnodal cubic. By Proposition
2.6, inverting in o maps the ellipse to an acnodal circular cubic γ , with o becoming
the isolated point of the cubic. The lines through o now parametrise the cubic, and
this parametrisation gives the same group on γ as the line construction that we gave
in Sect. 3.1 (see [22, Sect. 3.7]).
Concentric circles We now define a group on the union of two disjoint circles. For
notational convenience, we identify R2 with C. After an appropriate inversion, we can
assume the circles to be
with r > 1, and we represent each element of σ1 ∪ σ2 as r εe2πit with ε ∈ Z2 (with
the obvious convention r 0 = 1 and r 1 = r ). We define a group operation on σ1 ∪ σ2
by
r ε1 e2πit1 ⊕ r ε2 e2πit2 = r (ε1+ε2) mod 2 e2πi(t1+t2),
which turns σ1 ∪ σ2 into a group isomorphic to R/Z × Z2, with identity element
o = 1 = r 0e2πi·0. We again have the following concyclicity property, which is easily
seen using symmetry.
Proposition 3.4 Points a, b ∈ σ1 and c, d ∈ σ2 lie on a generalised circle if and only
if a ⊕ b ⊕ c ⊕ d = o. If a = b or c = d, then the generalised circle is tangent at that
point.
4 Constructions
4.1 Ellipse
Let σ be the ellipse defined by x 2 + (y/s)2 = 1, with the group structure introduced
in Sect. 3.2. Let n 5. We have a finite subgroup of size n given by
S =
k = 0, . . . , n − 1
By Proposition 3.3, the circle through any three points a, b, c ∈ S passes through the
point d = a b c ∈ S. Therefore, the only way S spans an ordinary circle is
when d coincides with one of the points a, b, c (which occurs if the circle is tangent
to σ at that point). It follows that the number of ordinary circles is equal to
(k1, k2, k3) ∈ Zn3
2k1 + k2 + k3 ≡ 0 (mod n), k1, k2, k3 distinct ,
which is n2/2 − O(n).
Similarly, the number of 4point circles is equal to
(k1, k2, k3, k4) ∈ Zn4
k1 + k2 + k3 + k4 ≡ 0 (mod n), k1, k2, k3, k4 distinct ,
which is, by inclusion–exclusion, equal to (n3 − 6n2 + (8 + 3δn)n − 6εn )/24, where
δn is the number of solutions in Zn to the equation 2k = 0 and εn is the number of
solutions in Zn to the equation 4k = 0. This works out to
if n ≡ 1, 3 (mod 4),
4.2 Circular Cubic Curve
Let γ be an irreducible circular cubic, and let ⊕ be the group operation defined in
Sect. 3.1. It is well known (see for instance [13]) that the group (γ ∗, ⊕) is isomorphic
to the circle R/Z if γ is acnodal or if γ is smooth and has one connected component,
and is isomorphic to R/Z × Z2 if γ is smooth and has two connected components.
Let Hn be a subgroup of order n of γ ∗, and let x ∈ γ ∗ be such that 4x = ω h for
some h ∈ Hn. By Proposition 3.2, the number of ordinary generalised circles in the
coset S = Hn ⊕ x equals
2a ⊕ b ⊕ c = h, a, b, c distinct ,
2a ⊕ b ⊕ c = h, a, b, c =
x and distinct ,
which also equals n2/2 − O(n).
As in the previous construction, if o ∈/ S (equivalently, x ∈/ Hn) then the number of
4point circles is equal to (n3 − 6n2 + (8 + 3δn)n − 6εn)/24, where δn is the number
of solutions in Hn to the equation 2k = h and εn is the number of solutions in Hn to
the equation 4k = h. If Hn is cyclic, then we get the same numbers as in the previous
construction. Otherwise, n = 0 (mod 4), Hn =∼ Zn/2 × Z2, and the number of 4point
circles equals
which is greater than the corresponding number in the previous construction.
4.3 ‘Aligned’ Double Polygons
k2 ≡ k3 (mod m).
Let n 6 be even and set m = n/2. We identify R2 with C. Let σ1 be the
circle with centre the origin and radius one, and σ2 the circle with centre the
origin and radius r > 1. Let S1 = e2πik/m k = 0, . . . , m − 1 ⊂ σ1 and
S2 = r e2πik/m k = 0, . . . , m − 1 ⊂ σ2. Thus, S1 and S2 are the vertex sets of
regular mgons on σ1 and σ2 that are ‘aligned’ in the sense that their points lie at the
same set of angles from the common centre (see Fig. 4).
Let S = S1 ∪ S2. By Proposition 3.4, the points a, b ∈ σ1, c, d ∈ σ2 are collinear or
concyclic if and only if a ⊕ b ⊕ c ⊕ d = o. In particular, if a = b, then the generalised
circle through the three points is tangent to σ1. It follows that if n 8, the ordinary
generalised circles of S are exactly those through e2πik1/m , r e−2πik2/m , r e−2πik3/m
or through r e−2πik1/m , e2πik2/m , e2πik3/m where 2k1 + k2 + k3 ≡ 0 (mod m), with
For generic r > 1, we then obtain that the number of ordinary generalised circles
equals
2k1 + k2 + k3 ≡ 0 (mod n), k2, k3 distinct
(although k2 and k3 are not ordered, we either have two points on σ1 or two points on
σ2). This equals m(m − 2) if m is even and m(m − 1) if m is odd. That is, for generic
r , we obtain n2/4 − n ordinary generalised circles if n ≡ 0 (mod 4) and n2/4 − n/2
ordinary generalised circles if n ≡ 2 (mod 4).
If we choose r = (cos(2π k/m))−1 (there are m/4 choices for r ), then the tangent
lines at points of S1 pass through two points of S2, so are ordinary generalised circles.
Thus, for these choices of r we lose m ordinary circles, and obtain n2/4−3n/2 ordinary
circles if n ≡ 0 (mod 4) and n2/4 − n ordinary circles if n ≡ 2 (mod 4). Note that
this is much less than the number of ordinary circles given by Constructions 4.1 and
4.2.
Similarly, the number of 4point generalised circles spanned by S equals
k1 + k2 + k3 + k4 ≡ 0 (mod m), k1 = k2 and k3 = k4 ,
which is m3/4 − O(m2) = n3/32 − O(n2), also much less than the number in
Constructions 4.1 and 4.2.
4.4 ‘Offset’ Double Polygons
We modify the previous construction by rotating S2 around the origin by an angle
of π k/m. This results in S2 = r e−iπ(2k−1)/m k = 0, . . . , m − 1 and S =
S1 ∪ S2 (see Fig. 5). As before, if n 8, the ordinary generalised circles of
S are exactly those through e2πik1/m , r e−iπ(2k2−1)/m , r e−iπ(2k3−1)/m or through
r e−iπ(2k1−1)/m , e2πik2/m , e2πik3/m , where 2k1 + k2 + k3 ≡ 1 (mod m) with k2 ≡ k3
(mod m).
Fig. 5 ‘Offset’ double hexagon
For generic r > 1, we now have to count the number of ordered triples in the set
2k1 + k2 + k3 ≡ 1 (mod n), k2, k3 distinct .
This equals m2 if m is even and m(m − 1) if m is odd. That is, for generic r , we obtain
n2/4 ordinary generalised circles if n ≡ 0 (mod 4), worse than Construction 4.3, and
n2/4 − n/2 ordinary generalised circles if n ≡ 2 (mod 4), the same number as in
Construction 4.3.
Again, if we choose r = (cos(2π k/m))−1 (there are m/4 choices for r ), we lose
m ordinary circles. Thus, we obtain n2/4 − n ordinary circles if n ≡ 2 (mod 4), the
same number as in Construction 4.3.
As in Construction 4.3, we get n3/32 − O(n2) 4point circles.
4.5 Punctured Double Polygons
ordinary generalised circles. That is, there are 3n2/8 − n + 5/8 ordinary generalised
circles if n ≡ 1 (mod 4).
Next assume that m is even. Before we remove p, there are m(m − 2) ordinary
generalised circles, of which there are (m − 2)/2 through two different points of S2
tangent at p, and there are also m − 2 ordinary generalised circles through p tangent
at a point of S2. As before, we obtain
ordinary generalised circles. Thus, we obtain 3n2/8−3n/2+17/8 ordinary generalised
circles if n ≡ 3 (mod 4).
Instead of starting with Construction 4.3, we can take the ‘offset’ Construction 4.4
and remove a point. It is easy to see that when n ≡ 1 (mod 4) we obtain the same
number of ordinary generalised circles, while if n ≡ 3 (mod 4) we obtain more.
Since there are no 5point circles in Constructions 4.3 and 4.4 when m 6,
removing a point does not add any 4point circle, but destroys O(n2) of them. We thus
get n3/32 − O(n2) 4point generalised circles, which is asymptotically the same as
in Constructions 4.3 and 4.4.
4.6 Inverted Double Polygons
We can use inversion to make new constructions out of old ones.
Invert Construction 4.5 in the removed point p. The resulting point set has m points
on a circle and m − 1 points on a line disjoint from the circle. Every ordinary circle
after the inversion corresponds to an ordinary generalised circle not passing through
p before the inversion. If m is odd, there are (m − 1)/2 ordinary generalised circles
tangent at p and a further m − 1 ordinary generalised circles through p tangent to σ2,
so we obtain m(m − 1) − 3(m − 1)/2 = (m − 1)(2m − 3)/2 ordinary circles. For
even m we similarly obtain m(m − 2) − 3(m − 2)/2 = (m − 2)(2m − 3)/2 ordinary
circles. That is, we have (n − 1)(n − 2)/4 = n2/4 − 3n/4 + 1/2 ordinary circles when
n ≡ 1 (mod 4) and (n − 3)(n − 2)/4 = n2/4 − 5n/4 + 3/2 ordinary circles when
n ≡ 3 (mod 4).
If we remove another point from this inverted construction, we obtain a set of n
points where n is even, with 3n2/8 − O (n) ordinary circles.
4.7 Other Inverted Examples
If we invert Construction 4.1 in a point on the ellipse that is not in the set S, then by
Proposition 2.6, we obtain points on an acnodal circular cubic (without its acnode) as
in Construction 4.2, with the same number of ordinary and 4point generalised circles.
If we invert a circular cubic in a point not on the curve, then we obtain a bicircular
quartic by Lemma 2.5. There will again be n2/2 − O (n) ordinary circles (or ordinary
generalised circles) and n3/24 − O (n2) 4point circles among the inverted points.
5 The Structure Theorems
The proofs of our structure theorems for sets with few ordinary circles crucially rely
on the following structure theorem for sets with few ordinary lines due to Green and
Tao [13]. Recall that an ordinary line is a line containing exactly two points of the
given point set.
Theorem 5.1 (Green–Tao) Let K > 0 and let n be sufficiently large depending on K .
If a set P of n points in R2 spans at most K n ordinary lines, then P differs in at most
O (K ) points from an example of one of the following types:
(i) n − O (K ) points on a line;
(ii) m points each on a line and a disjoint conic, for some m = n/2 ± O (K );
(iii) n ± O (K ) points on an acnodal or smooth cubic.
We commence the proof of Theorem 1.4. Let P be a set of n points spanning at most
K n2 ordinary generalised circles. We wish to show that P lies mostly on a bicircular
quartic (we will repeatedly use ‘mostly’ to mean ‘for all but O (K ) points’).
Note that for at least 2n/3 points p of P , there are at most 9K n ordinary circles
through p, hence the set I p( P \{ p}) spans at most 9K n ordinary lines. Let P be the
set of such points. For n sufficiently large depending on K , applying Theorem 5.1 to
I p( P \{ p}) for any p ∈ P gives that I p( P \{ p}) lies mostly on a line, a line and a
conic, an acnodal cubic, or a smooth cubic.
If there exists p ∈ P such that I p( P \{ p}) lies mostly on a line, then inverting
again in p, we see that P must lie mostly on a line or a circle.
If there exists p ∈ P such that I p( P \{ p}) lies mostly on a line and a disjoint
conic σ , we have two cases, depending on whether p lies on or not.
If p ∈ , we invert again in p to find that P lies mostly on the union of and
I p(σ ). By Lemma 2.5, I p(σ ) is either a circle (if σ is a circle) or an irreducible
bicircular quartic (if σ is a noncircular conic). Furthermore, p is the only point that
could possibly lie on both and I p(σ ). Since roughly n/2 points of P lie on , there
must be another point q ∈ ∩ P that does not lie on I p(σ ). In Iq ( P \{q}), the line
remains a line, and by definition of P the set Iq ( P \{q}) spans few ordinary lines,
so Theorem 5.1 tells us Iq (I p(σ )) is a conic. It follows from Lemma 2.5 that I p(σ )
cannot be a quartic, since we inverted in the point q outside I p(σ ) and did not obtain
a quartic. That means I p(σ ) has to be a circle, and it is disjoint from . Thus, P lies
mostly on the union of a line and a disjoint circle.
If p ∈/ , we invert in p to see that P lies mostly on the union of the circle I p( )
and the curve I p(σ ), which is either a circle or a quartic. Again p is the only point that
can lie on both curves. Inverting in another point q ∈ I p( ) ∩ P , Iq (I p( )) becomes
a line, so Theorem 5.1 tells us that Iq (I p(σ )) is a conic, so that I p(σ ) must be a circle
disjoint from I p( ) as before. Thus, P lies mostly on the union of two disjoint circles.
The case that remains is when for all p ∈ P , the set I p( P \{ p}) lies mostly on an
acnodal or smooth cubic γ . Fix such a p, and consider I p(γ ), which mostly contains
P . If γ is not a circular cubic, then by the classification in Sect. 2 it has circular degree
three, so I p(γ ) has circular degree three as well. For any q ∈ I p(γ ) ∩ P other than
p, the curve Iq (I p(γ )) is also a cubic curve, by the definition of P and Theorem 5.1.
By Case (iii) of Lemma 2.5, this can only happen if q is a singularity of I p(γ ). But
I p(γ ) is an irreducible curve of degree at most six, and so has at most ten singularities
by [23, Thm. 4.4], which is a contradiction. So γ must be a circular cubic that is
acnodal or smooth. If γ is acnodal, then I p(γ ) is either a bicircular quartic (if p ∈/ γ ),
an acnodal circular cubic (if p is a regular point of γ ), or a noncircular conic (if p is
the singularity of γ ). In the last case, the conic is an ellipse by Proposition 2.6. If γ is
smooth, then I p(γ ) is either a bicircular quartic or a smooth circular cubic.
We have encountered the following curves that P could mostly lie on: a line, a
circle, an ellipse, a disjoint union of a line and a circle, a disjoint union of two circles,
a circular cubic, or a bicircular quartic. All of these are subsets of bicircular quartics,
which proves the statement of Theorem 1.4.
5.2 Proof of the Strong Structure Theorem
We now prove Theorem 1.5. First of all, as explained in Sect. 4, a subgroup of an
ellipse and an appropriate coset of a subgroup of a smooth circular cubic both have
at most n2/2 ordinary generalised circles, and a double polygon has at most n2/4
ordinary generalised circles. It follows from Lemma 5.2 below that if we add and/or
remove O (K ) points, then there will be at most O (K n2) ordinary generalised circles.
Lemma 5.2 Let S be a set of n points in R2 with s ordinary generalised circles. Let
T be a set that differs from S in at most K points: S T  K . Then T has at most
s + O (K n2 + K 2n + K 3) ordinary generalised circles.
n
2 +
n + K − 1
2
= O (K n2 + K 2n + K 3)
ordinary generalised circles.
Next, let P be a set of n points with at most K n2 ordinary generalised circles. From
the proof of Theorem 1.4 above, we see that P differs in at most O (K ) points from a
line, a circle, an ellipse, a disjoint union of a line and a circle, a disjoint union of two
circles, a circular cubic, or a bicircular quartic. Moreover, in the proof we saw that
the circular cubic must be acnodal or smooth, and that the bicircular quartic has the
property that if we invert in a point on the curve, the resulting circular cubic is acnodal
or smooth.
Using inversions, we can reduce the number of types of curves that we need to
analyse further.
• If P lies mostly on a line, then we are in Case (i) of Theorem 1.5, so we are done.
• If P lies mostly on a circle, then inverting in a point on the circle puts us in Case (i)
again.
• If P lies mostly on an ellipse, then inverting in a point of the ellipse places P
mostly on an acnodal circular cubic.
• If P lies mostly on a bicircular quartic, then inverting in any regular point on
the curve gives us a circular cubic. As mentioned above, this cubic is acnodal or
smooth.
• If P lies mostly on a line and a disjoint circle, then an inversion in a point not on
the line or circle places P mostly on two disjoint circles.
• If P lies mostly on the disjoint union of two circles, we can apply an inversion
that maps the two disjoint circles to two concentric circles [6, Thm. 1.7].
So, up to inversions, we need only consider the cases when P lies mostly on
an acnodal or smooth circular cubic, or on two concentric circles. We do this in
Lemmas 5.5 and 5.6 below, which will complete the proof of Theorem 1.5.
To determine the structure of P , we use a variant of a lemma from additive
combinatorics that was used by Green and Tao [13]. It captures the principle that if a finite
subset of a group is almost closed under addition, then it is close to a subgroup. The
following statement is Proposition A.5 in [13].
Proposition 5.3 Let K > 0 and let n be sufficiently large depending on K . Let
A, B, C be three subsets of some abelian group (G, ⊕), all of cardinality within K of
n. Suppose there are at most K n pairs (a, b) ∈ A × B for which a ⊕ b ∈/ C . Then
there is a subgroup H G and cosets H ⊕ x , H ⊕ y such that
(H ⊕ x ), B
(H ⊕ y), C
(H ⊕ x ⊕ y) = O(K ).
The variant that we need is a simple corollary of Proposition 5.3.
Corollary 5.4 Let K > 0 and let n be sufficiently large depending on K . Let
A, B, C, D be four subsets of some abelian group (G, ⊕), all of cardinality within
K of n. Suppose there are at most K n2 triples (a, b, c) ∈ A × B × C for which
a ⊕ b ⊕ c ∈/ D. Then there is a subgroup H G and cosets H ⊕ x , H ⊕ y, H ⊕ z
such that
(H ⊕ x ), B
(H ⊕ y), C
(H ⊕ z), D
(H ⊕ x ⊕ y ⊕ z) = O(K ).
Proof By the pigeonhole principle, there exists an a0 ∈ A such that there are at most
K n (where K = O(K )) pairs (b, c) ∈ B × C for which a0 ⊕ b ⊕ c ∈/ D, or
equivalently b ⊕ c ∈/ D a0. Applying Proposition 5.3, we have a subgroup H G
and cosets H ⊕ y, H ⊕ z such that
(H ⊕ y), C
(H ⊕ z), (D
(H ⊕ y ⊕ z) = O(K ).
Since B ∩ (H ⊕ y) n − O(K ), we repeat the argument above to obtain b0 ∈
B ∩(H ⊕ y) such that there are at most O(K n) pairs (a, c) ∈ A×C with a⊕b0⊕c ∈/ D,
and Proposition 5.3 gives a subgroup H G and cosets H ⊕ x , H ⊕ z such that
(H ⊕ x ), C
(H ⊕ z ), (D
(H ⊕ x ⊕ z ) = O(K ).
From this, it follows that (H ⊕z) (H ⊕z ) = O(K ), hence (H ⊕z)∩(H ⊕z )
n − O(K ). Since (H ⊕ z) ∩ (H ⊕ z ) is not empty, it has to be a coset of H ∩ H .
If H = H , then H ∩ H  n/2 + O(K ), a contradiction. Therefore, H = H and
H ⊕ z = H ⊕ z . So we have  A (H ⊕ x ), B (H ⊕ y), C (H ⊕ z), D (H ⊕
x ⊕ b0 ⊕ z) = O(K ). Since b0 ∈ H ⊕ y, we obtain D (H ⊕ x ⊕ y ⊕ z) = O(K )
as well.
Lemma 5.5 (Circular cubic) Let K > 0 and let n be sufficiently large depending on
K . Suppose P is a set of n points in R2 spanning at most K n2 ordinary generalised
circles, and all but at most K points of P lie on an acnodal or smooth circular cubic
γ . Then there is a coset H ⊕ x of a subgroup H γ ∗, with 4x ∈ H ⊕ ω, such that
 P (H ⊕ x ) = O(K ).
Proof Let P = P ∩ γ ∗. Then  P P  = O(K ), and by Lemma 5.2, P spans at
most O(K n2) ordinary circles. If a, b, c ∈ γ are distinct, then by Proposition 3.2, the
generalised circle through a, b, c meets γ again in the unique point d = ω (a ⊕b⊕c).
This implies that d ∈ P for all but at most O(K n2) triples a, b, c ∈ P , or equivalently
a ⊕ b ⊕ c ∈ ω P . Applying Corollary 5.4 with A = B = C = P and D = ω P ,
we obtain H γ ∗ and a coset H ⊕ x such that  P (H ⊕ x ) = O(K ) and
(ω P ) (H ⊕ 3x ) = O(K ), which is equivalent to  P (H 3x ⊕ ω) = O(K ).
Thus we have (H ⊕ x ) (H 3x ⊕ ω) = O(K ), which implies 4x ∈ H ⊕ ω.
Lemma 5.6 (Concentric circles) Let K > 0 and let n be sufficiently large depending
on K . Suppose P is a set of n points in R2 spanning at most K n2 ordinary generalised
circles. Suppose all but at most K of the points of P lie on two concentric circles, and
that P has n/2 ± O(K ) points on each. Then, up to similarity, P differs in at most
O(K ) points from an ‘aligned’ or ‘offset’ double polygon.
Proof By scaling and rotating, we can assume that P lies mostly on the two concentric
circles e2πit t ∈ [0, 1) and r e−2πit t ∈ [0, 1) , r > 1, which we gave a group
structure in Sect. 3.2.
Let P1 = P ∩ σ1 and P2 = P ∩ σ2. Then  P ( P1 ∪ P2) = O(K ), and by
Lemma 5.2, P1 ∪ P2 spans at most O(K n2) ordinary circles. If a, b ∈ σ1 and c ∈ σ2
with a = b, then by Lemma 3.4, the generalised circle through a, b, c meets σ1 ∪ σ2
again in the unique point d = (a ⊕ b ⊕ c). This implies d ∈ P2 for all but at most
O(K n2) triples (a, b, c) with a, b ∈ P1 and c ∈ P2. Applying Corollary 5.4 with
A = B = P1, C = P2 and D = P2, we get cosets H ⊕ x and H ⊕ y of σ1 ∪ σ2 such
that  P1 (H ⊕ x ),  P2 (H ⊕ y) = O(K ) and 2x ⊕ 2y ∈ H , where x ∈ σ1 and
y ∈ σ2. It follows that H σ1, hence H is a cyclic group of order m = n/2 ± O(K ),
and H ⊕ x and H ⊕ y are the vertex sets of regular mgons inscribed in σ1 and σ2,
respectively, either ‘aligned’ or ‘offset’ depending on whether x ⊕ y ∈ H or not.
Together these lemmas prove Theorem 1.5. It just remains to remark that if P differs
in O(K ) points from a coset on an acnodal circular cubic, then we apply inversion
in its singularity. By Proposition 2.6, we obtain that P differs in O(K ) points from a
coset H ⊕ x of a finite subgroup H of an ellipse, where 4x = o. Thus, x is a point
of the ellipse with eccentric angle a multiple of π/2. After a rotation, we can assume
that x = o, which is Case (ii) of Theorem 1.5.
6 Extremal Configurations
Suppose P is an npoint set in R2 spanning fewer than n2/2 ordinary generalised
circles, and that P is not contained in a generalised circle. Applying Theorem 1.5, we
can conclude that, up to inversions, P differs in O(1) points from one of the following
examples: points on a line, a coset of a subgroup of an acnodal or smooth circular
cubic, or a double polygon.
The first type of set is very easy to handle. Note that the lower bound is on the
number of ordinary circles, not counting 3point lines.
Lemma 6.1 Let K 1 and n 2K + 4. If all except K points of a set P ⊂ R2 of n
points lie on a line, then P spans at least n−1 ordinary circles.
2
Proof Let be a line such that  P ∩  = n − K . For any p ∈ P ∩ and q ∈ P\
there are at most K − 1 nonordinary circles through p, q, another point on P ∩ ,
and another point in P\ . Therefore, there are at least K (n − 2K ) ordinary circles
through p. This holds for any of the n − K points p ∈ P ∩ , and we obtain at least
K (n − 2K )(n − K )/2 ordinary circles. It is easy to see that when 1 K (n − 4)/2,
K (n − 2K )(n − K )/2 is minimised when K = 1.
Cosets on cubics are also relatively easy to handle. We again obtain a lower bound
on the number of ordinary circles, not including 3point lines.
Lemma 6.2 Suppose P ⊂ R2 differs in K points from a coset H ⊕ x of an acnodal
or smooth circular cubic, where H  = n ± O(K ) and 4x ω ∈ H . Then P spans at
least n2/2 − O(K n) ordinary circles.
Proof Suppose that P differs in K points from H ⊕ x . We know from Construction 4.2
that H ⊕ x spans n2/2 − O(n) ordinary circles, all of which are tangent to γ . We show
that adding or removing K points destroys no more than O(K n) of these ordinary
circles, so that the resulting set P still spans at least n2/2 − O(K n) ordinary circles.
Suppose we add a point q ∈/ H ⊕ x . For p ∈ H ⊕ x , at most one circle tangent to
γ at p can pass through q. Thus, adding q destroys at most n ordinary circles. Now
suppose we remove a point p ∈ H ⊕ x . Since ordinary circles of H ⊕ x correspond
to solutions of 2 p ⊕ q ⊕ r = ω or p ⊕ 2q ⊕ r = ω, there are at most O(n) solutions
for a fixed p. Thus removing p destroys at most O(n) ordinary circles.
Repeating K times, we see that adding or removing K points to or from H ⊕ x
destroys at most O(K n) ordinary generalised circles out of the n2/2 − O(n) spanned
by H ⊕ x . This proves that P spans at least n2/2 − O(K n) ordinary circles.
From the two lemmas above we know that there is an absolute constant C such that
a set of n points, not all collinear or concyclic, spanning at most n2/2 − C n ordinary
generalised circles, differs in O(1) points from Case (iv) in Theorem 1.5. This case,
where P is close to the vertex set of a double polygon, requires a more careful analysis
of the effect of adding or removing points.
We use the following special case of a result due to Raz et al. [20].
Proposition 6.3 If P ⊂ R2 is a set of n points contained in two circles, then the
number of lines with at least three points of P is at most O(n11/6).
Proof Denote the two circles by σ1 and σ2. We use [20, Thm. 6.1], which states that for
(not necessarily distinct) algebraic curves C1, C2, C3 of constant degree, and finite sets
Si ⊂ Ci , the number of collinear triples ( p1, p2, p3) ∈ S1 × S2 × S3, with p1, p2, p3
distinct, is bounded by O(S11/2S22/3S32/3 + S1 + S11/2S2 + S11/2S3),
unless C1 ∪ C2 ∪ C3 is a line or a cubic. Let C1 = σ1 and C2 = C3 = σ2. Set
Si = P ∩ Ci for i = 1, 2, 3. Every line with at least one point of S1 and two points
of S2 = S3 corresponds to a collinear triple in S1 × S2 × S3. Since the union of two
circles is not a line or a cubic, we can apply the theorem to get the bound O(n11/6)
for the number of collinear triples in P with one point in σ1 and two points in σ2.
Similarly, the number of collinear triples in P with one point in σ2 and two points
in σ1 is also O(n11/6). Since a line intersects σ1 ∪ σ2 in at most four points, we also
obtain the bound O(n11/6) for the number of lines with at least three points.
Lemma 6.4 Let S be a double polygon with m points on each circle. Let P = (S\ A)
∪ B be a set of n points, where A is a subset of S with a = O(1) points and B is a
set disjoint from S with b = O(1) points. Then P spans at least (2 + a + 4b)n2/8 −
O(n11/6) ordinary generalised circles.
Proof We know from Constructions 4.3 and 4.4 that S spans n2/4 − O(n) ordinary
generalised circles.
Consider first the number of ordinary generalised circles spanned by S\ A. As we
saw in Construction 4.5, removing a point p ∈ S destroys at most 3m/2 ordinary
generalised circles spanned by S, and adds m2/2 − O(m) = n2/8 − O(n) ordinary
generalised circles. Noting that there are at most m 4point generalised circles spanned
by S that go through any two given points of A, we thus have by inclusion–exclusion
that S\ A determines at least (1/4 + a/8)n2 − O(n) ordinary generalised circles.
Now consider adding q ∈ B to S. For any pair of points from S\ A, adding q ∈ B
creates a new ordinary generalised circle, unless the generalised circle through the
pair and q contains three or four points of S\ A. We already saw that the number of
ordinary generalised circles hitting a fixed point is O(n), so it remains to bound the
number of 4point generalised circles of S that hit q. If q lies on one of the concentric
circles, then no 4point generalised circles hit q, so we can assume that q does not.
Applying inversion in q reduces the problem to bounding the number of 4point lines
determined by a subset of two circles. By Proposition 6.3, this number is bounded by
O(n11/6), so p lies on at most O(n11/6) of the 4point generalised circles spanned
by S. Adding q to S thus creates at least n2 − O(n11/6) ordinary generalised circles.
Note that each p ∈ A that was removed destroys at most n of these circles.
Adding q to S\ A also destroys at most O(n) ordinary circles, since for each p ∈ S
there is only one circle tangent at p and going through q, and for each p ∈ A, at most
m ordinary circles spanned by S\ A go through p. Finally, since there are at most 2m
circles through two points of B that also go through two points of S\ A, P = (S\ A)∪ B
spans at least (1/4 + a/8 + b/2)n2 − O(n11/6) ordinary generalised circles.
Theorem 1.2 then follows easily from the lemmas above.
Proof of Theorem 1.2 Suppose that P is a set of n points in R2 with fewer than
n2/2 − C n ordinary generalised circles, where C is sufficiently large. Without loss of
generality, n is also sufficiently large. By Lemmas 6.1 and 6.2, we need only consider
the case where P differs by O(1) points from a double polygon. In the notation of
Lemma 6.4, we have P = (S\ A) ∪ B and (2 + a + 4b)/8 < 1/2, which implies that
a 1 and b = 0. So P is either equal to S, or is obtained from S by removing one
point, which are exactly the cases in Constructions 4.3, 4.4, and 4.5 . In particular,
the minimum number of ordinary generalised circles occurs in Construction 4.3 when
n ≡ 0 (mod 4), in Construction 4.5 when n ≡ 1, 3 (mod 4), and in Constructions 4.3
and 4.4 when n ≡ 2 (mod 4).
6.2 Ordinary Circles We now consider what happens if we do not count generalised circles that are lines, and prove Theorem 1.1.
Proof of Theorem 1.1 Let P be a set of n points not all on a line or a circle, with at most
n2/2 − C n ordinary circles, for a sufficiently large C . By a simple double counting
argument, there are at most n2/6 3point lines, so there are at most 2n2/3 − O(n)
ordinary generalised circles. By Theorem 1.5, up to inversions and up to O(1) points,
P lies on a line, an ellipse, a smooth circular cubic, or two concentric circles. By
Lemmas 6.1 and 6.2, the first three cases give us at least n2/2 − O(n) ordinary circles,
contrary to assumption. Therefore, we only need to consider the case where, when P
is transformed by an inversion to P , we have P = (S\ A) ∪ B, where S is a double
polygon (‘aligned’ or ‘offset’), and  A = a, B = b.
By Lemma 6.4, P has at least (2 + a + 4b)n2/8 − O(n11/6) ordinary generalised
circles, which gives us the inequality (2 + a + 4b)/8 < 2/3, which in turn gives us
a 3 and b = 0. Therefore, P lies on two concentric circles, and P lies on the
disjoint union of two circles or the disjoint union of a line and a circle.
Suppose that a = 3 (and b = 0). Then P has 5n2/8 − O(n) ordinary generalised
circles. Those passing through the centre of the inversion that transforms P to P , are
inverted back to straight lines passing through three points of P. As in the proof of
Lemma 6.4, there are n2/8 − O(n) ordinary generalised circles that pass through any
point of A. Also, we can use Lemma 6.5 below to show that there are at most O(n)
ordinary generalised circles spanned by S\ A that intersect in the same point not in S.
Indeed, by Lemma 6.5, there are at most n/2 ordinary generalised circles of S that
intersect in the same point p ∈/ S. Furthermore, for each point q ∈ A there are O(n)
generalised circles through p, q, and two more points of S. It follows that there are
O(n) ordinary generalised circles spanned by S\ A through p.
Thus, if the centre of inversion is in A, P has n2/2 − O(n) ordinary circles, which
is a contradiction if C is chosen large enough. On the other hand, if the centre of
inversion is not in A, then P has 5n2/8 − O(n) ordinary circles, also a contradiction.
Therefore, we have a 2, which means that P is a set of n points as in
Constructions 4.3, 4.4, 4.5, or 4.6.
Next, suppose that n is even. If a = 2, then there are n2/2 − O(n) ordinary
generalised circles and through both points of A there are n2/8 − O(n) ordinary
generalised circles. If we invert in one of these points in A, we obtain a set with 3n2/8−
O(n) ordinary circles (as in Construction 4.6), which is not extremal. Otherwise,
a = 0, P is as in Constructions 4.3 or 4.4, and there are at least n2/4 − n ordinary
generalised circles if n ≡ 0 (mod 4) and n2/4 − n/2 if n ≡ 2 (mod 4). Let p be
the centre of the inversion that transforms P to P . Then all the 3point lines of P
are inverted to ordinary circles in the double polygon P , all passing through p. By
Lemma 6.5 below, there are at most n/2 ordinary circles that intersect in the same point
not in P . Thus, in P there at most n/2 3point lines, and the number of ordinary circles
(not including lines) is at least n2/4 − 3n/2 if n ≡ 0 (mod 4) and n2/4 − n if n ≡ 2
(mod 4), which match Construction 4.3 (and Construction 4.4 if n ≡ 2 (mod 4)), if
the radii are chosen so that each vertex of the inner polygon has an ordinary generalised
circle that is a straight line tangent to it.
Finally, suppose that n is odd. Then a = 1 and P is as in Construction 4.5,
with 3n2/8 − O(n) ordinary generalised circles. It follows that P must be as in
Construction 4.6, with n2/4 − 3n/4 + 1/2 ordinary circles if n ≡ 1 (mod 4) and
n2/4 − 5n/4 + 3/2 ordinary circles if n ≡ 3 (mod 4). This finishes the proof.
Lemma 6.5 Let S be a double polygon (‘aligned’ or ‘offset’) with m points on each
circle. Then a point q ∈/ S lies on at most m ordinary generalised circles spanned
by S.
Proof Denote the inner circle by σ1 and the outer circle by σ2, both with centre o. We
proceed by case analysis on the position of q with respect to σ1 and σ2. Note that for
each point p ∈ S, at most one of the ordinary generalised circles tangent at p can go
through q.
If q lies on either σ1 or σ2, then q does not lie on any ordinary generalised circle
spanned by S.
If q lies inside σ1, then q lies on at most m ordinary generalised circles spanned by
S, since ordinary generalised circles tangent to σ1 cannot pass through q. Similarly,
if q lies outside σ2, it lies on at most m ordinary generalised circles, since ordinary
generalised circles tangent to σ2 lie inside σ2.
The remaining case to consider is when q lies in the annulus bounded by σ1 and σ2.
Consider the subset S ⊂ S of points p such that there exists an ordinary generalised
circle tangent at p going through q. Consider the four circles passing through q and
tangent to both σ1 and σ2. They touch σ1 at a1, b1, c1, d1 and σ2 at a2, b2, c2, d2 as
in Fig. 6. Any circle through q tangent to σ1 and intersecting σ2 in two points, must
touch σ1 on one of the open arcs a1b1 or c1d1. Similarly, any circle through q tangent
to σ2 and intersecting σ1 in two points, must touch σ2 on one of the open arcs a2c2 or
b2d2. It follows that S must be contained in the relative interiors of one of these four
arcs. Since S consists of m equally spaced points on each of σ1 and σ2,
where θ and ϕ are as indicated in Fig. 6. In order to show that S  m, it suffices to
show that the angle sum θ + ϕ is strictly less than π . This is clear from Fig. 6 (note
that a1, o, a2 are collinear with a1 and a2 on opposite sides of o).
6.3 FourPoint Circles
Proof of Theorem 1.3 Let P be a set of n points in R2 with at least n3/24 − 7n2/24 +
O(n) 4point generalised circles. Let ti denote the number of i point lines (i 2) and
si the number of i point circles (i 3) in P. By counting unordered triples of points,
we have
n
3 =
n3 − O(n2)
and t3 + s3 = O(n2), so we can apply Theorem 1.5. We next consider each of the
cases of that theorem in turn.
If all except O(1) points of P lie on a straight line, it is easy to see that P determines
only O(n2) generalised circles, contrary to assumption.
If all except O(1) are vertices of two regular mgons on concentric circles where
m = n/2± O(1), then we know from Constructions 4.3, 4.4, and 4.5 that P determines
at most n3/32 + O(n2) 4point generalised circles, again contrary to assumption.
Suppose next that P = ((H ⊕ x )\ A) ∪ B, where H is a finite subgroup of order
m = n ± O(1) of a smooth circular cubic, A is a subset of H ⊕ x with a = O(1)
points, and B is a set disjoint from H ⊕ x with b = O(1) points. Then n = m − a + b.
The number of 4point generalised circles in H ⊕ x is m3/24 − m2/4 + O(m). We
next determine an upper bound for the number of 4point generalised circles in P.
For each p ∈ A, let C p be the set of 4point generalised circles of H ⊕ x that pass
through p. Then C p = m2/6 − O(m) and C p ∩ Cq  = O(m) for distinct p, q ∈ A.
By inclusion–exclusion, we destroy at least p∈A C p am2/6 − O(m) 4point
generalised circles by removing A, and we still have at most m3/24−m2/4−am2/6+
O(m) 4point generalised circles in (H ⊕ x )\ A.
For each p ∈ B, the number of ordinary generalised circles spanned by H ⊕ x
passing through p is at most O(m). This is because each such generalised circle is
tangent to the cubic at one of the points of H ⊕ x , and there is only one generalised
circle through p and tangent at a given point of H ⊕ x . Also, for each pair of distinct
p, q ∈ B, there are at most O (m) generalised circles through p and q and two points
of H ⊕ x ; and for any three p, q, r ∈ B there are at most O (1) generalised circles
through p, q, r and one point of H ⊕ x . Therefore, again by inclusion–exclusion, by
adding B we gain at most O (m) 4point generalised circles.
It follows that the number of 4point generalised circles determined by P is
am2 + O (m) =
n3 − (a + 3b + 6)n2 + O (n)
Since we assumed that
n3 − 7n2 + O (n)
we obtain a + 3b < 1. Therefore, a = b = 0 and P = H ⊕ x . The maximum number
of 4point circles in a coset has been determined in Constructions 4.1 and 4.2.
The final case, when all but O (1) points of P lie on an ellipse, can be reduced to the
previous case. Indeed, by Lemma 2.6, if we invert the ellipse in a point on the ellipse,
we obtain an acnodal circular cubic, and then the above analysis holds verbatim for
the group of regular points on this cubic.
Acknowledgements M. Makhul and J. Schicho were supported by the Austrian Science Fund (FWF):
DKW1214, subproject DK9. H.N. Mojarrad and F. de Zeeuw were partially supported by Swiss National
Science Foundation grants 200020165977 and 200021162884.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0
International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit to the original author(s) and the
source, provide a link to the Creative Commons license, and indicate if changes were made.
1. Bálintová , A. , Bálint , V. : On the number of circles determined by n points in the Euclidean plane . Acta Math. Hung . 63 ( 3 ), 283  289 ( 1994 )
2. Ball , S. : On sets defining few ordinary planes . arXiv:1606. 02138 ( 2016 )
3. Ball , S. , Monserrat , J. : A generalisation of Sylvester's problem to higher dimensions . J. Geom. doi:10. 1007/s0002201603578
4. Basset , A.B.: An Elementary Treatise on Cubic and Quartic Curves . Cambridge Deighton Bell, London ( 1901 )
5. Berger , M. , Pansu , R. , Berry , J.P. , SaintRaymond , X. : Problems in Geometry . Problem Books in Mathematics. Springer, New York ( 1984 )
6. Blair , D.E. : Inversion Theory and Conformal Mappings. Student Mathematical Library, vol. 9. American Mathematical Society, Providence ( 2000 )
7. Boys , T. , Valculescu , C., de Zeeuw , F. : On the number of ordinary conics . SIAM J . Discrete Math. 30 ( 3 ), 1644  1659 ( 2016 )
8. Brass , P. , Moser , W. , Pach , J. : Research Problems in Discrete Geometry. Springer, New York ( 2005 )
9. Coxeter , H.S.M. : Introduction to Geometry . Wiley Classics Library. Wiley, New York ( 1969 )
10. Czaplin´ski, A. , Dumnicki , M. , Farnik , Ł. , Gwoz´dziewicz, J. , LampaBaczyn ´ska, M. , Malara , G. , Szemberg , T. , Szpond , J. , TutajGasin ´ska, H.: On the SylvesterGallai theorem for conics . Rend. Sem. Mat. Univ. Padova (in press) . arXiv:1411.2648
11. Eisenbud , D. , Green , M. , Harris , J. : CayleyBacharach theorems and conjectures . Bull. Am. Math. Soc. (N .S.) 33 ( 3 ), 295  324 ( 1996 )
12. Elliott , P.D. T .A.: On the number of circles determined by n points . Acta Math. Acad. Sci. Hung . 18 ( 12 ), 181  188 ( 1967 )
13. Green , B. , Tao, T.: On sets defining few ordinary lines . Discrete Comput. Geom . 50 ( 2 ), 409  468 ( 2013 )
14. Hilton , H. : Plane Algebraic Curves . The Clarendon Press, Oxford ( 1920 )
15. Joachimsthal , F. : Démonstration d'un théorème de Mr . Steiner . J. Reine Angew. Math. 36 , 95  96 ( 1848 )
16. Johnson , W.W. : Classification of plane curves with reference to inversion . The Analyst 4, 42  47 ( 1877 )
17. Klee , V. , Wagon , S. : Old and New Unsolved Problems in Plane Geometry and Number Theory . The Dolciani Mathematical Expositions, vol. 11. Mathematical Association of America , Washington, DC ( 1991 )
18. Motzkin , T. : The lines and planes connecting the points of a finite set . Trans. Am. Math. Soc . 70 ( 3 ), 451  464 ( 1951 )
19. Nassajian Mojarrad , H. , de Zeeuw , F. : On the number of ordinary circles . arXiv:1412. 8314 ( 2014 )
20. Raz , O.E. , Sharir , M. , de Zeeuw , F. : Polynomials vanishing on Cartesian products: The ElekesSzabó theorem revisited . Duke Math. J . 165 (18), 3517  3566 ( 2016 )
21. Silverman , J.H.: The Arithmetic of Elliptic Curves . Graduate Texts in Mathematics , vol. 106 , 2nd edn. Springer, Dordrecht ( 2009 )
22. Silverman , J.H., Tate , J. : Rational Points on Elliptic Curves. Undergraduate Texts in Mathematics, 2nd edn . Springer, New York ( 1992 )
23. Walker , R.J. : Algebraic Curves. Springer, New York ( 1978 )
24. Werner , T.R. : Rational Families of Circles and Bicircular Quartics. Ph.D. Thesis , FriedrichAlexanderUniversität ErlangenNürnberg ( 2011 ). https://opus4.kobv.de/opus4fau/files/2301/ ThomasWernerDissertation.pdf
25. Wiseman , J.A. , Wilson , P.R. : A Sylvester theorem for conic sections . Discrete Comput. Geom . 3 ( 4 ), 295  305 ( 1988 )
26. Zhang , R.: On the number of ordinary circles determined by n points . Discrete Comput. Geom . 46 ( 2 ), 205  211 ( 2011 )