Rectification Principles in Additive Number Theory
Discrete Comput Geom
Y. F. Bilu 2 3
V. F. Lev 1 3
I. Z. Ruzsa 0 3
0 Mathematical Institute of the Hungarian Academy of Sciences , Pf. 127, H-1364 Budapest , Hungary
1 Department of Mathematics, University of Georgia , Athens, GA 30602 , USA
2 Forschungsinstitut fu ̈r Mathematik, ETH-Zentrum , CH-8092 Zurich , Switzerland
3 Rectification Principles in Additive Number Theory
We consider two general principles which allow us to reduce certain additive problems for residue classes modulo a prime to the corresponding problems for integers. ¤ Research of the first author was supported by the SFB 170 “Geometrie und Analysis” at Go¨ttingen. Research of the third author was supported by Hungarian National Foundation for Scientific Research, Grant No. 17433.
It is well known that additive problems in finite abelian groups are generally more difficult
than analogous problems in Z. For instance, consider the following classical problem:
given an abelian group G, describe all pairs of finite sets K ; L µ G such that
jK C L j < jK j C jL j:
When G D Z (or a torsion-free abelian group) the answer is almost trivial: K and L
must be arithmetic progressions with the same difference. When G is a cyclic group of
prime order, the answer is given by Vosper’s theorem [
], which is quite nontrivial. And
when G is an arbitrary finite abelian group, we should turn to the extremely complicated
recursive classification of Kemperman [
]. (A few years ago the problem was solved
for torsion-free nonabelian groups [
Nevertheless, more than 30 years ago, Freiman [10, Sect. 3.13] discovered that, at
least for cyclic groups of prime order, certain additive problems can be reduced to
corresponding problems in Z, provided the sets in consideration are “not very large.”
Seemingly, this important observation did not receive much attention.
On the other hand, it was recently observed in [
] that there is another reduction
method. It is weaker than Freiman’s method in that it requires the sets to be “very
small” instead of “not very large.” However, unlike the method of Freiman it imposes
no additional restrictions on the sets and handles easily the case of distinct summands,
which makes it applicable when Freiman’s method fails.
The objective of this paper is to apply Freiman’s discovery and the reduction method
mentioned above to concrete additive problems.
2. Freiman’s Rectification Principle
For simplicity, we consider only the case of equal summands: K D L. Using [22,
Lemma 3.3] we can extend the results to distinct summands.
We need the concept of Fs -isomorphism [
]. Let G; H be abelian groups, and
consider subsets K ½ G and L ½ H. The bijection ': K ! L is Freiman’s isomorphism
of order s or, shortly, Fs -isomorphism, if for any a1; : : : ; a2s 2 K ,
a1 C ¢ ¢ ¢ C as D asC1 C ¢ ¢ ¢ C a2s
if and only if
'.a1/ C ¢ ¢ ¢ C '.as / D '.asC1/ C ¢ ¢ ¢ C '.a2s /:
It is easily seen that, if K and L are FsC1-isomorphic, then they are also Fs -isomorphic.
Clearly, two sets K and L are F1-isomorphic if and only if jK j D jLj.
Theorem 2.1 (Freiman’s Rectification Principle). For any positive numbers ¾ 2 R
and s 2 Z there exists a positive constant c1 D c1.¾; s/ such that the following holds:
Let p be a prime number and let K µ Z= pZ satisfy jK j · c1 p and
jK C K j < ¾ jK j:
Then there exists a set of integers K 0 ½ Z such that the canonical homomorphism
Z ! Z= pZ induces an Fs -isomorphism of K 0 onto K .
To put it briefly, this theorem asserts that if a set of residues has a small sumset and
is not too large itself, then it is Fs -isomorphic to a set of integers.
A proof of Theorem 2.1 (for s D 2) is briefly sketched in [10, Sect. 3.12]. Our proof
given below is simpler than the original, but requires substantially no new ideas.
The argument is based on the following result of Freiman:
Theorem 2.2 (Freiman). Let ¾ be a positive real number and let K be a finite set of
jK C K j < ¾ jK j:
Then there exist positive integers r; b1; : : : ; br and nonzero integers g0; g1; : : : ; gr such
r · c2.¾ /;
b1 ¢ ¢ ¢ br · c3.¾ /jK j;
K µ 5
D 5.g0I g1; : : : ; gr I b1; : : : ; br /
:D fg0 C ¯1g1 C ¢ ¢ ¢ C ¯r gr : ¯i D 0; : : : ; bi ¡ 1g;
where c2.¾ / and c3.¾ / are positive constants, depending only on ¾ .
There are two different proofs of this theorem. The first is Freiman’s original, see [
], and [
]. The second is due to Ruzsa, see [
] and [
Proof of Theorem 2.1.
We shall see that the theorem holds with
c1 D .2sc2.2¾ //¡c2.2¾ /.c3.2¾ //¡1:
Let K0 µ f0; 1; : : : ; p ¡ 1g be the preimage of K under the canonical homomorphism
Z ! Z= pZ. Then jK0 C K0j · 2jK C K j < 2¾ jK0j, whence by Theorem 2.2 we have
K0 µ 5.g0I g1; : : : ; gr I b1; : : : ; br /, where r; b1; : : : ; br are positive integers satisfying
r · c2.2¾ /;
b1 ¢ ¢ ¢ br · c3.2¾ /jK j;
and g1; : : : ; gr 2 Z.
Put "i D .2sr bi /¡1. Then
p"1 ¢ ¢ ¢ "r D p.2sr /¡r .b1 ¢ ¢ ¢ br /¡1
¸ p.2sc2.2¾ //¡c2.2¾ /.c3.2¾ //¡1jK j¡1 D c1 pjK j¡1 ¸ 1:
Hence by Minkowski’s theorem on linear inequalities [7, App. B, Theorem III], there
exists a nonzero vector .a; a1; : : : ;ar / 2 ZrC1 such that
jaj < p;
¡ ai ¯¯ · "i
where k ¢ k stands for the distance from the nearest integer. This gives
°° a.x ¡ g0/ °°
°° p °° ·
r ° agi °°
X bi °°° p °° · .2s/¡1
for any x 2 K0.
½ ¹ p º
¡ .2s/ ; ¡
¹ p º
C 1; : : : ;
¹ p º¾
be the set of integers congruent modulo p to one of the numbers a.x ¡ g0/ (for some
x 2 K0). Clearly, K1 is mapped onto K by x 7! ux C g0 mod p, where u is an arbitrary
integer satisfying au ´ 1 .mod p/. It follows that K1 is Fs -isomorphic to K , for any
algebraic sum of 2s elements of K1 which is 0 modulo p is also 0 in Z.
Finally, we define K 0 D fux C g0: x 2 K1g.
Freiman’s rectification principle allows us to reduce various additive problems in
Z= pZ to corresponding problems in Z. Unfortunately, we have to make the restrictive
assumption jK j · c1 p. Restricting to “not very large” sets is the price we have to pay
for the use of such a powerful tool as Freiman’s Theorem 2.2. For instance, the main
results of [
] and [
] also assume that the sets in question are small enough.
It would be nice to find a proof of the rectification principle independent of
Theorem 2.2. For ¾ < 2:4 and s D 2 such a proof is implicit in [10, Sect. 2.3], where the
following result is obtained.
Theorem 2.3 [
]. Let K µ Z= pZ satisfy jK CK j < 2:4jK j, and suppose, in addition,
that jK j < p=35. Then K can be covered by a “short” arithmetic progression modulo
p: There exist g0; g1 2 Z= pZ and a positive integer b · jK C K j ¡ jK j C 1 such that
K µ fg0 C g1¯: ¯ D 0; : : : ;b ¡ 1g:
Conjecturally, the assertion of this theorem is true for jK CK j · maxf p¡1; 3jK j¡4g.
Using Theorem 2.1, we can easily prove it for sufficiently small K .
Indeed, suppose that jK j < c1 p, where c1 D c1.3; 2/ is the constant of Theorem 2.1.
Let K 0 ½ Z be the set of integers which is F2-isomorphic to K and is mapped onto K
by the canonical mapping Z ! Z= pZ. Then K 0 also satisfies jK 0 C K 0j · 3jK 0j ¡ 4
(since jK 0j D jK j and jK 0 C K 0j D jK C K j by the definition of an Fs -isomorphism).
By another and well-known result of Freiman [10, Theorem 1.9] (which has a relatively
easy elementary proof; see also [
]), there exist a; d 2 Z and a positive integer b ·
jK 0 C K 0j ¡ jK 0j C 1 such that
K 0 µ fa C d¯: ¯ D 0; : : : ;b ¡ 1g:
Now (2.2) is clearly satisfied if g0 and g1 are the elements of Z= pZ congruent to a and
This is a first illustrative example which shows how rectification methods can be used
to reduce difficult additive problems in Z= pZ to easier problems in Z.
3. Direct Rectification
Theorem 2.1 shows that any (not too large) set of residues K µ Z= pZ with a small sumset
is isomorphic to a set of integers. It turns out that this is true for any K , regardless of
its sumset, provided that jK j is very small. Specifically, it is shown in [
] that any
K µ Z= pZ of the cardinality k D jK j is contained in an arithmetic progression modulo
p of at most
terms. If k · log4 p C log4 log4 p (where log4 is the logarithm base 4), then the
number (3.1) is less than p=2 C 1, whence K is F2-isomorphic to a set of integers. Similarly,
if k · log2s p C log2s log2s p, then K is Fs -isomorphic to a set of integers. Essentially
the same can be obtained by a direct application of the idea we used in the proof of
Theorem 3.1. Let K µ Z= pZ, where p is a prime. If jK j · log2s p, then there exists
a set of integers K 0 ½ Z such that the canonical homomorphism Z ! Z= pZ induces
an Fs -isomorphism of K 0 onto K .
Proof. Let K D fg1; : : : ; gr g, and put "1 D ¢ ¢ ¢ D "r D .2s/¡1. Since r · log2s p, we
have p"1 ¢ ¢ ¢ "r ¸ 1. Therefore, applying Minkowski’s theorem exactly in the same way
as in the proof of Theorem 2.1, we find a 2 Z satisfying
a 6´ 0 .mod p/;
°° agi °°
°° p °° · .2s/¡1
Now let mi 2 f¡b p=.2s/c; ¡b p=.2s/c C 1; : : : ; b p=.2s/cg be defined from mi ´
agi .mod p/, and put K 0 D fum1; : : : ; umk g where u is any integer, inverse to a in
Z= pZ. Then the canonical homomorphism Z ! Z= pZ maps K 0 onto K , and this
mapping is an Fs -isomorphism, which follows immediately from jmi j < p=.2s/ (as in the
proof of Theorem 2.1).
This theorem is nearly best possible: here is an example which shows that there exists
a set K µ Z= pZ of cardinality jK j · 2 log2 p C 1 which is not F2-isomorphic to any
set of integers. (This example can easily be generalized to produce a set of cardinality
at most 2 logs p C 1 which is not Fs -isomorphic to any set of integers.)
Put N D blog2 pc, and write p D 2d1 C 2d2 C ¢ ¢ ¢ C 2dt , where 0 · d1 < d2 < ¢ ¢ ¢ <
dt ; t · N C 1. We define
K D f0g [ f1; 2; 4; : : : ; 2N g [ f2d1 C 2d2 ; 2d1 C 2d2 C 2d3 ; : : : ; 2d1 C 2d2 C ¢ ¢ ¢ C 2dt¡1 g
(all the numbers are considered as residues modulo p), so that jK j · 2N C 1. We assume
that K is F2-isomorphic to a set of integers
K 0 D f0g [ fa0; a1; a2; : : : ; aN g [ fb2; b3; : : : ; bt¡1g
and then from
which yields subsequently
and obtain a contradiction. Let ': K ! K 0 be the isomorphism. As the notation suggests,
we suppose (which does not restrict the generality) that '.0/ D 0 and that ai ; bi are the
images in K 0 of the corresponding elements of K . Then for any i 2 [0; N ¡ 1], the
equality in Z= pZ
0 C 2iC1 D 2i C 2i
0 C aiC1 D ai C ai ;
for i D 2; : : : ; t ¡ 2 we obtain
But this is a contradiction in view of
a1 D 2a0;
a2 D 4a0 ; : : : ;
aN D 2N a0:
0 C .2d1 C 2d2 / D 2d1 C 2d2
b2 D .2d1 C 2d2 /a0;
0 C .2d1 C ¢ ¢ ¢ C 2di C 2diC1 / D .2d1 C ¢ ¢ ¢ C 2di / C 2diC1
biC1 D bi C adiC1 ;
b3 D .2d1 C 2d2 C 2d3 /a0; : : : ;
bt¡1 D .2d1 C 2d2 C ¢ ¢ ¢ C 2dt¡1 /a0:
To show how Theorem 3.1 can be applied in the case of distinct summands, consider
the following problem: Given s sets K1; : : : ; Ks in an abelian group G, how many
representations of the form
x D a1 C ¢ ¢ ¢ C as I ai 2 Ki
.i D 1; : : : ; s/
can an element x 2 G have? We assume here that the cardinalities jKi j are preassigned.
For G D Z, we have the following result:
Theorem 3.2 [16, Theorem 1]. Let K1; : : : ; Ks µ Z be a finite sets of integers. Then
the number of solutions of (3.2) is maximized, when x D 0, and Ki are the sets of
Ki D f®i ; ®i C 1; : : : ; °i g
.i D 1; : : : ; s/;
where integers ®i ; °i are chosen in such a way that
°i ¡ ®i C 1 D jKi j;
j®i C °i j · 1
.i D 1; : : : ; s/;
j.®1 C °1/ C ¢ ¢ ¢ C .®s C °s /j · 1:
Using direct rectification we can easily transfer Theorem 3.2 to small subsets of
Theorem 3.3. Let K1; : : : ; Ks µ Z= pZ be sets of residues modulo a prime p, and
assume that jK1j C ¢ ¢ ¢ C jKs j · log2s p. Then the number of solutions of (3.2) is
maximized, when x D 0, and Ki are the sets of consecutive residues
Ki D f®i ; ®i C 1; : : : ; °i g .mod p/
.i D 1; : : : ; s/;
where the integers ®i ; °i are chosen in such a way that
°i ¡ ®i C 1 D jKi j;
j®i C °i j · 1
.i D 1; : : : ; s/;
j.®1 C °1/ C ¢ ¢ ¢ C .®s C °s /j · 1:
Proof. For any abelian group G and for any L1; : : : ; Ls µ G, x 2 G, denote by
Nx .L1; : : : ; Ls / the number of solutions of
a1 C ¢ ¢ ¢ C as D x I
ai 2 Li ;
N .L1; : : : ; Ls / D mx2aGx Nx .L1; : : : ; Ls /:
Define K D K1 [ ¢ ¢ ¢ [ Ks . Let ': K 0 ! K be an Fs -isomorphism of a set of integers
K 0 µ Z onto K , and let Ki0 be the preimage of Ki in K 0 .i D 1; : : : ; s/. Then evidently
with ai0; as0Ci 2 Ki0 holds if and only if
holds, and it follows that
a10 C ¢ ¢ ¢ C as0 D as0C1 C ¢ ¢ ¢ C a20s
'.a10/ C ¢ ¢ ¢ C '.as0 / D '.as0C1/ C ¢ ¢ ¢ C '.a20s /
N .K1; : : : ; Ks / D N .K 10; : : : ; Ks0 /:
By Theorem 3.2, the right-hand side can only increase if, for all i D 1; : : : ; s, we replace
Ki0 by Ki00 D f®i ; ®i C 1; : : : ; °i g µ Z, where ®i ; °i satisfy (3.3) and (3.4):
N .K 10; : : : ; Ks0 / · N .K 100; : : : ; Ks00/:
Now, let K i be the images of Ki00 under the canonical homomorphism Z ! Z= pZ.
The assertion of the theorem follows from the observation that for any integer x 2
f¡b p=2c; ¡b p=2c C 1; : : : ; b p=2cg with the corresponding residue x 2 Z= pZ,
Nx .K 100; : : : ; Ks00/ D Nx .K 1; : : : ; K s /;
N .K 100; : : : ; Ks00/ D N .K 1; : : : ; K s /:
Notice, that in the proof above we could not apply Theorem 2.1 not only because
the sets Ki are distinct, but also (and mainly) because there are no restrictions on the
cardinalities j2Ki j. However, Theorem 3.1 works perfectly in this situation.
4. Erdo˝s–Heilbronn Conjecture
Let h ¸ 2 be an integer and let K be a subset of the set of elements of an abelian group.
Denote by h b K the set of all sums of h distinct elements from K :
h b K D fa1 C ¢ ¢ ¢ C ah : a1; : : : ; ah 2 K and ai 6D aj for 1 · i < j · hg:
Let p be a prime. Erdo˝s and Heilbronn (see [9, p. 95]) conjectured that any K µ Z= pZ
j2b K j ¸ minf2jK j ¡ 3; pg:
Note that the inequality j2b K j ¸ 2jK j ¡ 3 trivially holds for K µ Z (and for finite
subsets of torsion-free abelian groups). In general, we have
Proposition 4.1 (Folklore).
(a) For any positive integer h and any finite set K µ Z we have
jh b K j ¸ hjK j ¡ h2 C 1:
(b) If jK j ¸ maxfh C2; 5g, then equality in (4.2) holds if and only if K is an arithmetic
For a proof see [19, Theorems 1 and 2], [5, App. C], or [20, Theorem 1.10].
After a number of partial results, say, [
], and [
] (see [
] for more
references), the Erdo˝s–Heilbronn conjecture (4.1) was finally settled by Dias da Silva and
]. Another proof was suggested in [
]. Actually, Dias da Silva and
Hamidoune proved a more general inequality
jh b K j ¸ minfhjK j ¡ h2 C 1; pg
for arbitrary h ¸ 2 and K µ Z= pZ.
Recently Alon et al. [
] obtained a fairly general additive theorem which contains
inequality (4.3) as a particular case.
However, to the best of our knowledge, the problem of when the equality in (4.3)
holds is still open. Here we obtain an answer for sufficiently small K µ Z= pZ as a
direct consequence of Proposition 4.1(b) and Theorem 2.1.
Theorem 4.1. For any h ¸ 2 there exists a constant c4 D c4.h/ with the following
property. For any prime p and any set of residues K µ Z= pZ such that
maxfh C 2; 5g · jK j · c4 p;
jh b K j D hjK j ¡ h2 C 1
holds if and only if K is an arithmetic progression.
Proof. Put c4.h/ D minfh¡1; c1.2h; h/g, where c1 is defined in Theorem 2.1. If K µ
Z= pZ is an arithmetic progression and jK j · p= h, then (4.5) obviously holds.
Conversely, assume that K µ Z= pZ satisfies (4.4) and (4.5). Fix an .h ¡ 2/-element
subset H µ K and denote L D K nH . Then j2b Lj · jh b K j < hjK j. Therefore
jK C K j · j2b K jCjK j · j2b LjCjH C K jCjK j < hjK jC.h ¡2/jK jCjK j < 2hjK j:
By Theorem 2.1, the set K is Fh -isomorphic to a set of integers K 0 µ Z. Then clearly
jh b K 0j D jh b K j D hjK 0j ¡ h2 C 1, and by Proposition 4.1(b), K 0 is an arithmetic
progression. Then so is K .
Freiman et al. [
] applied a similar “rectification” approach for h D 2. Their
technique is quite different and is not based on Theorem 2.2, and for h D 2 their result is
much stronger than Theorem 4.1 above. However, the method of [
] does not extend
to h ¸ 3. See also Ro¨dseth [
When the Erdo˝s–Heilbronn conjecture was proved, it had been conjectured by the
second author that in fact a much more general result holds. Specifically, let K and L
be subsets of an abelian group, such that jK j · jLj, and let ¿ : K ! L be an arbitrary
mapping from K to L. Define K C L to be the set of all the sums a C b (where
a 2 K ; b 2 L) such that b 6D ¿ .a/:
K C L D fa C b: a 2 K ; b 2 L ; and b 6D ¿ .a/g:
Conjecture 4.1 (Lev). Let K and L be subsets of Z= pZ satisfying jK j · jLj, and let
¿ : K ! L be an arbitrary mapping from K to L. Then
jK C Lj ¸ minfjK j C jLj ¡ 3; pg:
Using Theorems 2.1 or 3.1 we are able to prove this for small K ; L. First, we need a
corresponding result in Z.
Theorem 4.2. Let K and L be finite subsets of Z satisfying jK j · jLj, and let ¿ : K !
L be an arbitrary mapping from K to L. Then
jK C Lj ¸ jK j C jLj ¡ 3:
Proof. Write down the elements of K and L in ascending order: K D fa1; : : : ; ak g and
L D fb1; : : : ; bl g, where ai < aj and bi < bj for i < j .
We first assume that jK j < jLj. Then there exists bj 2 L which is not an image of an
element of K under ¿ . Therefore among the k C l ¡ 1 distinct sums
at most one sum in the first row and at most one sum in the last row are excluded by the
condition b 6D ¿ .a/. At least k C l ¡ 3 remaining sums fall into K C L.
Now assume jK j D jLj. Then either there exists bj 2 L which has no preimage in
K , and we can repeat the argument above; or ¿ is a bijection, in which case we consider
k C l ¡ 1 distinct sums
a1 C b1 < a1 C b2 < ¢ ¢ ¢ < a1 C bl < a2 C bl < a3 C bl < ¢ ¢ ¢ < ak C bl ;
and observe again, that at most two of them may not fall into K C L.
Theorem 4.3. The assertion of Conjecture 4.1 holds provided that either L D K and
jK j D jLj · c5 p (with a sufficiently small absolute constant c5), or jK j C jLj · log4 p.
Proof. In the first case (L D K ; jK j D jLj · c5 p) we observe that
jK C K j · jK C K j C jK j · 3jK j ¡ 4;
assuming jK C K j < 2jK j ¡ 3. Then by Theorem 2.1, K is F2-isomorphic to a
set of integers K 0. Let ¿ 0: K 0 ! K 0 be the mapping induced by ¿ . Then K 0 satisfies
jK 0 C K 0j < 2jK 0j ¡ 3, which, as Theorem 4.2 shows, is impossible for K 0 µ Z.
In the second case (jK j C jLj · log4 p), we find, as in Theorem 3.3, a set of integers
M µ Z which is F2-isomorphic to the union K [ L, define K 0; L0 µ M to be the
preimages of K ; L ; respectively, and define ¿ 0: K 0 ! L0 to be the mapping induced by
¿ . Then by Theorem 4.2,
jK C Lj D jK 0 C L0j ¸ jK 0j C jL0j ¡ 3 D jK j C jLj ¡ 3:
Using [22, Lemma 3.3] the last theorem can be extended to all sets K and L such that
"jLj · jK j · jLj · c6."/ p for any " > 0.
As a concluding remark, we note that the rectification method can be used not only
for the group Z= pZ: for instance, in [
] it is applied for the torus Rm =Zm .
We would like to thank Dani Berend and the referee for pointing out some inaccuracies
in the manuscript.
1. N. Alon , M. B. Nathanson , and I. Z. Ruzsa , Adding distinct congruence classes modulo a prime , Amer. Math. Monthly , 102 ( 1995 ), 250 - 255 .
2. N. Alon , M. B. Nathanson , and I. Z. Ruzsa , The polynomial method and restricted sums of congruence classes , J. Number Theory , 56 ( 1996 ), 404 - 417 .
3. Yu . Bilu, Structure of sets with small sumsets, Mathe´matiques Stochastiques , Univ. Bordeaux 2 , Preprint 94-10, Bordeaux, 1994 .
4. Yu . Bilu, The .® C 2¯/-inequality on the torus, J. London Math. Soc . To appear.
5. Yu . Bilu, Addition of sets of integers of positive density , J. Number Theory . To appear.
6. L. V. Brailovski and G. A. Freiman , On a product of finite subsets in a torsion-free group , J. Algebra , 130 ( 1990 ), 462 - 476 .
7. J. W. S. Cassels , An Introduction to Diophantine Approximations, Cambridge Tracts in Mathematics and Mathematical Physics , vol. 45 , Cambridge University Press, Cambridge, 1965 .
8. J. A. Dias da Silva and Y. O. Hamidoune , Cyclic spaces for Grassmann derivatives and additive theory , Bull. London Math. Soc. , 26 ( 1994 ), 140 - 146 .
9. P. Erdo ˝s and R. L. Graham , Old and New Problems and Results in Combinatorial Number Theory, L 'Enseignement Mathe´matique, Geneva, 1980 .
10. G. A. Freiman , Foundations of a Structural Theory of Set Addition (Russian) , Kazan' , 1966 ; English translation: Translation of Mathematical Monographs , vol. 37 , American Mathematical Society, Providence, RI, 1973 .
11. G. A. Freiman , What is the structure of K if K C K is small? , In: Number Theory, New York 1984-1985 (D. V. Chudnovsky et al., eds.) , Lecture Notes in Mathematics , vol. 1240 , Springer-Verlag, New York, 1987 , pp. 109 - 134 .
12. G. A. Freiman , L. Low , and J. Pitman, The proof of Paul Erdo˝s' conjecture of the addition of different residue classes modulo a prime number , In: Structure Theory of Set Addition , preprint, Tel Aviv-Marseilles, 1992 = 93 .
13. Y. O. Hamidoune , An isoperimetric method in additive theory , J. Algebra , 179 ( 1996 ), 622 - 630 .
14. I. H. B. Kemperman , On small sumsets in an abelian group , Acta Math. , 103 ( 1960 ), 63 - 88 .
15. V. F. Lev , Simultaneous approximations and covering by arithmetic progressions in Fp . In preparation.
16. V. F. Lev , On the number of solutions of a linear equation over finite sets . Submitted.
17. V. F. Lev and P. Yu . Smeliansky, On addition of two distinct sets of integers, Acta Arith ., LXX.1 ( 1995 ), 85 - 91 .
18. R. Mansfield , How many slopes in a polygon? Israel J . Math., 39 ( 1981 ), 265 - 272 .
19. M. B. Nathanson , Inverse theorems for subset sums , Trans. Amer. Math. Soc. , 347 ( 1995 ), 1409 - 1418 .
20. M. B. Nathanson , Additive Number Theory: 2. Inverse Theorems and the Geometry of Sumsets , Graduate Texts in Mathematics, vol. 165 , Springer-Verlag, New York, 1996 .
21. O ¨. J. Ro ¨dseth, Sums of distinct residues mod p, Acta Arith ., LXV.2 ( 1993 ), 181 - 184 .
22. I. Z. Ruzsa , Arithmetical progressions and the number of sums , Period Math. Hungar., 25 ( 1 ) ( 1992 ), 105 - 111 .
23. I. Z. Ruzsa , Generalized arithmetical progressions and sumsets , Acta Math. Hungar. , 65 ( 4 ) ( 1994 ), 379 - 388 .
24. A. G. Vosper, The critical pairs of subsets of a group of prime order , J. London Math. Soc. , 31 ( 1956 ), 200 - 205 , 280 - 282 .