Covering Spheres with Spheres

Discrete & Computational Geometry, Sep 2007

Given a sphere of any radius r in an n-dimensional Euclidean space, we study the coverings of this sphere with solid spheres of radius one. Our goal is to design a covering of the lowest covering density, which defines the average number of solid spheres covering a point in a bigger sphere. For growing dimension n, we design a covering that gives the covering density of order (nln n)/2 for a sphere of any radius r>1 and a complete Euclidean space. This new upper bound reduces two times the order nln n established in the classic Rogers bound.

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Covering Spheres with Spheres

Discrete Comput Geom Covering Spheres with Spheres Ilya Dumer Given a sphere of any radius r in an n-dimensional Euclidean space, we study the coverings of this sphere with solid spheres of radius one. Our goal is to design a covering of the lowest covering density, which defines the average number of solid spheres covering a point in a bigger sphere. For growing dimension n, we design a covering that gives the covering density of order (n ln n)/2 for a sphere of any radius r > 1 and a complete Euclidean space. This new upper bound reduces two times the order n ln n established in the classic Rogers bound. Spherical coverings. Let Brn(x) be a ball (solid sphere) of radius r centered at some point x = (x1, . . . , xn) of an n-dimensional Euclidean space Rn: - 1 Introduction n i=1 We also use a simpler notation Brn if a ball is centered at the origin x = 0. For any subset A ⊆ Rn, we say that a subset Cov(A, ε) ⊆ Rn forms an ε-covering (an ε-net) of A if A is contained in the union of the balls of radius ε centered at points x ∈Cov(A, ε). In this case, we use notation Cov(A, ε) : A ⊆ Bεn(x). x∈Cov(A,ε) By changing the scale in Rn, we can always consider the rescaled set A/ε and the new covering Cov(A/ε, 1) with unit balls B1n(x). Without loss of generality, below we consider these (unit) coverings. One of the classical problems is to obtain tight bounds on the covering size |Cov(Brn, 1)| for any ball Brn of radius r and dimension n. Another related covering problem arises for a sphere Srn d=ef z ∈ Rn+1 zi2 = r2 . n+1 Then a unit ball B1n+1(x) intersects this sphere with a spherical cap Crn(ρ , y) = Srn ∩ B1n+1(x), which has some center y ∈ Srn, half-chord ρ ≤ 1, and the corresponding half-angle α = arcsin ρ/r . The biggest possible cap Cn(1, y) is obtained if the center x of the r corresponding ball B1n+1(x) is centered at the distance ( 1 ) ( 2 ) ( 3 ) x = from the origin. To obtain a minimal covering, we shall consider the biggest caps Crn(1, y) assuming that all the centers x satisfy ( 1 ). Covering density. Given a set A ⊆ Rn, let |A| denote n-dimensional volume (Lebesque measure) of A. We then consider any unit covering Cov(A, 1) and define minimum covering density min ϑ (A) = Cov(A,1) x∈Cov(A,1) |B1n(x) ∩ A| |A| . Minimal coverings have been long studied for the spheres Srn and the balls Brn. The celebrated Coxeter–Few–Rogers lower bound [1] shows that for a sufficiently large ball Brn, ϑ (Srn) ≥ c0 n. Here and below ci denote some universal constants. A similar result also holds for any sphere Srn of radius r ≥ n (see Example 6.3 in [4]). Various upper bounds on the minimum covering density are obtained for Brn and Srn by Rogers in the classic papers [2] and [3]. In particular, it follows from these papers that for a sufficiently large radius r , a ball Brn and a sphere Srn can be covered with density ϑ ≤ Despite recent improvements obtained in [4] and [5], respectively, for spheres Srn and balls Brn of a relatively small radius r , the Rogers bound ( 3 ) is still the best asymptotic bound known for sufficiently large spheres, balls, and complete spaces Rn of growing dimension n. For a sphere Srn of any dimension n ≥ 3 and an arbitrary radius r > 1, the best universal upper bound known to date is obtained in [4] (see Corollary 1.2 and Remark 5.1): ϑ (Srn) ≤ 2 1 + ln n Our main result is presented in Theorem 1, which reduces about two times the present upper bounds ( 3 ) and ( 4 ) for n → ∞. Theorem 1 Unit balls can cover a sphere Srn of any radius r > 1 and any dimension n ≥ 3 with density For n → ∞, there exists o( 1 ) → 0 such that ϑ (Srn) ≤ ϑ (Srn) ≤ The following corollary to Theorem 1 (see also [8]) shows that the Rogers bound can also be reduced about two times for the coverings of complete Euclidean spaces Rn. Corollary 2 For n → ∞, unit balls can cover the entire Euclidean space Rn with density ϑ (Rn) ≤ 1 2 + o( 1 ) n ln n. 2 Preliminaries: Embedded Coverings In this section, we obtain the estimates on ϑ (Srn) that are similar to ( 3 ) and ( 4 ). However, we will introduce here a slightly different technique of embedded coverings that will be substantially extended in Section 3 to improve the former bounds ( 3 ) and ( 4 ). We will also employ most of our calculations performed in this section. Consider a sphere Srn of some dimension n ≥ 3 and radius r > 1. We use notation C(ρ , y) for a cap Crn(ρ , y) whenever parameters n and r are fixed; we also use a shorter notation C(ρ) when a specific center y is of no importance. In this case, Cov(ρ) will denote any covering of Srn with spherical caps C(ρ). By definition, a covering Cov(ρ) has covering density where Ωρ is the fraction of the surface of the sphere Srn covered by a cap C(ρ), ϑρ = Ωρ |Cov(ρ)| Ωρ = |C(ρ)| |Srn| . ( 4 ) ( 5 ) ( 6 ) ( 7 ) For any τ < ρ ≤ 1, we extensively use inequality (see Corollary 3.2 (ii) in [4]): and its particular version Ωτ ≥ Ω1τ n obtained for ρ = 1. We begin with two preliminary lemmas, which will simplify our calculations. Let f1(x) and f2(x) be two positive differentiable functions. We say that f1(x) moderates f2(x) for x ≥ a if for all x ≥ a, Lemma 3 Consider m functions fi (x) such that f1(x) moderates each function fi (x), i ≥ 2, for x ≥ a. Then inequality Ωτ ≥ Ωρ τ n ρ f1(x) f2(x) f1(x) ≥ f2(x) . f1(x) ≥ fi (x) si (x) d=ef x fi (t ) a fi (t ) dt. m i=2 m i=2 ( 8 ) (9) (10) Proof For n = 4, . . . , 7, the above inequality is verified numerically. Let t = ln1n . We take the logarithm of the left part of (9) and use power series. This gives the following inequalities t −n ln 1 − n = ∞ i=1 t i ni−1i < t + t 2 ∞ i=1 Here we observe that t < 12 for n ≥ 8, and replace the left-hand side of the first inequality with geometric series in its right-hand side, which in turn is bounded by holds for any x ≥ a if it is valid for x = a. Proof Note that fi (x) = fi (a) exp{si (x)}, where Also, si (x) ≤ s1(x) for all i ≥ 2. Therefore, which completes the proof. Lemma 4 For any n ≥ 4, 1 1 − n ln n −n < 1 + 1/ln n + 1/ln2 n. m i=2 f1(x) ≥ fi (a) exp{s1(x)} ≥ fi (a) exp{si (x)} = fi (x), m i=2 taking n = 8. We also note that for any n ≥ 8, For any z < 12 we can also use inequality To design a covering Cov( 1 ), we also use another covering 1 ε = n ln n , λ = 1 + ρ = 1 − ε, with smaller caps C(ε, u). In our first step, we randomly choose N points y ∈Srn, Consider the set {C(ρ , y)} of N caps. In the second step, we take all centers u ∈ Cov(ε) that are left uncovered by the set {C(ρ , y)} and form the extended set This set {x} covers the entire set Cov(ε) and therefore forms a unit covering, if the caps C(ρ , x) are expanded to the caps C(1, x), Lemma 5 For any n ≥ 8, covering {x} has density n Cov( 1 ) : Sr ⊆ C(1, x). x∈{x} ϑ∗ ≤ Proof Any point u is covered by some cap C(ρ , y) with probability Ωρ . Our goal is to estimate the expected number N of centers u left uncovered after N trials. Here N = (1 − Ωρ )N · |Cov(ε)|. Note that 1 − Ωρ ≤ e−Ωρ (1+Ωρ/2). Also, according to estimate (15), N Ωρ > λn ln n − Ωρ . Thus, (1 − Ωρ )N ≤ e−(λn ln n−Ωρ )(1+Ωρ/2) ≤ e−λn ln n. Here the second inequality holds since Ωρ < Ω1 < 1/2, whereas λn ln n > 16. Next, we estimate the covering size |Cov(ε)|. Let this covering have some density ϑε. Then inequality ( 8 ) gives the estimates (17) (18) (19) (20) (21) |Cov(ε)| = ϑε/Ωε ≤ (n ln n)nϑε/Ω1, N ≤ e−λn ln n(n ln n)nϑε/Ω1 ≤ ϑε/(n2Ω1). According to our design, there exists a covering {x} with caps C(1, x) that has size at most N + N . Next, we combine last inequality with (15), and estimate the covering density of {x}: ϑ1 = Ω1(N + N ) ≤ (λn ln n)Ω1/Ωρ + ϑε/n2. Finally, we estimate Ω1/Ωρ using inequalities ( 8 ) and (9): Ω1/Ωρ ≤ (1 − ε)−n ≤ 1 + 1/ln n + 1/ln2 n. Thus, we can rewrite our estimate ϑ1 as using notation ϑ1 ≤ ϑ∗(1 − 1/n2) + ϑε/n2 ϑ∗ = λn ln n 1 + 1/ln n + 1/ln2 n 1 − 1/n2 . For any given n, bound (20) only depends on the density ϑε. Now let us assume that ϑ1 meets some known upper bound ϑ˜1 valid for all (or sufficiently large) radii r . For example, we can use ( 4 ) or any weaker estimate. Next, we can change the scale in Rn+1 and replace a covering Cov( 1 ) of a sphere Srn/ε with the covering Cov(ε) of the sphere Srn. This rescaling shows that we can replace ϑε in (20) with any known upper bound ϑ˜1. In turn, inequality (20) shows that the bound ϑ˜1 will be further reduced by this replacement as long as ϑ˜1 > ϑ∗. Thus, this iteration process yields the upper bound ϑ1 ≤ ϑ∗. Our final step is to verify that the upper bound ϑ∗ of (21) also satisfies inequality (16). Here we use Lemma 3. Then straightforward calculations show that the right-hand side of (16) exceeds the right-hand side of (21) for n = 8 and moderates it for n ≥ 8, due to the bigger remaining term 3/ ln n in (16). Remark More detailed arguments show that (16) holds for n ≥ 3, whereas for n ≥ 8, ϑ∗ ≤ Covering design. In this section, we obtain a covering of the sphere Srn with asymptotic density (n ln n)/2. The new design will use both the former covering Cov(ε) (with slightly different parameters) and another covering Cov(μ) with a larger radius μ that has asymptotic order of n−1/2. Namely, we use parameters and proceed as follows. A. Let a sphere Srn be covered with two different coverings Cov(μ) and Cov(ε): , Cov(μ) : Srn ⊆ Cov(ε) : Srn ⊆ z∈Cov(μ) We assume that both coverings have the former density ϑ∗ of (16) or less. B. Randomly choose N points y ∈ Srn and consider the corresponding spherical caps C(ρ , y), where C. Let C(μ, z¯) be any cap in Cov(μ) that contains at least one center u ∈ Cov(ε) not covered by the ρ-caps. We consider all such centers z¯ and form the joint set {x} = {y} ∪ {z¯}. This set covers Cov(ε) with ρ-caps and therefore forms the required covering Cov( 1 ) : Srn ⊆ C(1, x). x∈{x} We now proceed with preliminary discussion, which outlines the main steps of the proof. (22) (23) Outline of the proof. Let us first assume that we keep the design of Section 2 but apply it to the new covering Cov(μ) instead of Cov(ε). This will require taking ρ = 1 − μ to cover the centers of the caps C(μ, z) and then expanding ρ to 1 to cover the whole μ-caps. Contrary to our former choice of ρ = 1 − ε in (19), this expansion will exponentially increase the covering density. Namely, straightforward calculations show that Ω1/Ω1−ε → 1, Ω1/Ω1−μ = exp{n1/2}, To circumvent this problem, we keep ρ = 1 − ε in (22) but change our design as follows. 1. Given any cap C(μ, z), we say that a cap C(ρ , y) is d -close if y falls within distance d < ρ to z. In our proof, we refine the selection of the caps C(ρ , y) and count only d -close caps, instead of the ρ-close caps considered in Section 2. It is easy to verify that distance d of (22) is so close to ρ that (24) (25) (26) Ωρ /Ωd → 1, For this reason, counting only d -close caps instead of the former ρ-close caps will carry no overhead to the covering size (23). 2. On the other hand, we will show in Lemma 6 that the μ-cap becomes almost completely covered by a cap C(ρ , y) when the latter becomes d -close instead of ρ-close. Namely, only a vanishing fraction ω ≈ exp(− 32 ln2 n) of a μ-cap is left uncovered in this case. 3. We shall also use the fact that a typical μ-cap is covered by multiple d -close caps. According to (23), the average number Ωd N of these caps has the exact order of λn ln n: λn ln n − Ωd < Ωd N ≤ λn ln n. In our proof, we first define insufficiently covered μ-caps. Namely, we call a cap C(μ, z ) non-saturated if it has only s or fewer d -close caps, where s = n/q , q = 3 ln ln n. This choice of s will achieve two goals. 4. We prove in Lemma 7 that non-saturated caps typically form a very small fraction of order exp[−λn ln n] among all μ-caps. On the other hand, it is easy to see that the quantity |Cov(μ)| ≤ ϑ∗/Ωμ exceeds N by the exponential factor Ωd /Ωμ ∼ exp[βn ln n] or less. Then our choice of λ and β in (22) gives the expected number N = o(N ) of non-saturated caps. Thus, non-saturated caps typically form a vanishing fraction of not only μ-caps but also ρ-caps. 5. Next, we proceed with saturated μ-caps and count all those centers u ∈ Cov(ε) that are left uncovered by random ρ-caps. All caps C(μ, z ) that contain uncovered centers u are called porous. For a given s, we show in Lemma 8 that the set {u } forms a very small portion of Cov(ε), with expected fraction of ωs+1 ∼ exp[− 23 n ln n]. Note that the quantity |Cov(ε)| ≤ ϑ∗/Ωε exceeds N by the exponential factor Ωd /Ωε ∼ exp[n ln n] or less. Therefore, the expected size of {u } is N = o(N ). 6. Finally, the centers of all non-saturated and porous caps are combined into the set z¯ ={z , z }. Then the set {x} = {y, z¯} completely covers the set Cov(ε) with the caps C(ρ , x). Therefore, {x} also covers Srn with unit caps. Main proofs. To prove Theorem 1, we first observe (by numerical comparison) that the existing bound (16) is tighter for n ≤ 100 than bound ( 5 ) of Theorem 1 or even its refined version (39) considered below. Thus, Theorem 1 holds for n ≤ 100. For this reason, we shall only consider dimensions n ≥ 100. In the end of the proof, we also address the asymptotic case n → ∞, wherein the calculations are more straightforward. The proof is based on three lemmas. Consider two caps C(μ, Z) and C(ρ , Y) with centers Y and Z, which are d -close. These caps are represented in Fig. 1. Here the origin O is the center of Srn. Lemma 6 For any cap C(μ, Z), a randomly chosen d -close cap C(ρ , Y) fails to cover any given point x of C(μ, Z) with probability p(x) ≤ ω, where 1 ω = 4 ln n 1 3 ≤ 4 ln n exp − 2 ln2 n . (27) Proof The caps C(μ, Z) and C(ρ , Y) have bases PQRSA and PMRTB, which form the balls Bμn (A) and Bρn(B). Below we consider the boundary PQRS of C(μ, Z), which forms the sphere Sn−1(A). The bigger cap C(ρ , Y) covers this boundary, with μ the exception of the cap PQR centered at Q. We first consider the case, when x is a boundary point and belongs to PQRS. Then the probability p(x) is the fraction Ω of the entire boundary contained in uncovered cap PQR. We first estimate the half-angle α = PAQ formed by the cap PQR. Let d(H, G) denote the distance between any two points H and G. Also, let σ (H) be the distance from a point H to the line OBY that connects the origin O with the center B of the bigger base Bρn(B) and with the center Y of the cap C(ρ , Y). We use inequalities σ (A) ≤ σ (Z) ≤ d(Z, Y) ≤d. On the other hand, consider the base PNR of the uncovered cap PQR. Here N denotes the center of this base. Then both lines AN and BN are orthogonal to this base. Also, d(B, P) = ρ, and d(N, P) ≤ d(A, P) = μ. Thus, d(B, N) = d2(B, P) − d2(N, P) ≥ ρ2 − μ2 ≥ ρ − μ2. The latter inequality follows from the trivial inequality 2ρ − 1 ≥ μ2. Finally, the center line OBY is orthogonal to the entire base PMRTB and its line BN. Then d(B, N) = σ (N) and d(A, N) ≥ σ (N) − σ (A) ≥ ρ − μ2 − d ≥ ε. Now we consider the right triangle ANP and deduce that (28) (29) (30) cos α = d(A, N)/d(A, P) ≥ ε/μ = 3 n ln n. (Note that the latter expression exceeds 1 for n < 42, which simply implies that a d-close cap entirely covers μ-cap for small n.) Given α, we can estimate the fraction Ω of the boundary sphere Sn−1(A) contained in the uncovered cap PQR. This μ fraction can be defined as (see [4]) Ω < {2π(n − 1)}−1/2 sinn−1 α < 6π cos α n n − 1 −1/2 sinn−1 α ln n < sinn−1 α 4 ln n . approximation 1 − x ≤ e−x−x2/2. This gives inequalities 1 ) > 16. Now we use (30) and The last inequality follows from the fact that 6π(1 − n sinn−1 α ≤ 3 1 − n ln2 n ≤ exp − 32 ln2 n − 49n ln4 n Finally, consider the second case, when a point x does not belong to the boundary PQRS. Therefore, x is taken from a smaller cap C(μ , Z) ⊂ C(μ, Z) with the same center Z and a half-chord μ < μ. Similarly to the first case, we define the boundary Sn−1(A ) of C(μ , Z), where A is some center on the line AZ. Again, p(x) is the μ fraction Ω of this boundary left uncovered by the bigger cap C(ρ , Y). To obtain the upper bound on Ω , we only need to replace μ with a smaller μ in (30). This gives the smaller angle α ≤ arccos(ε/μ ), which is reduced to 0 if ε ≥ μ . In particular, the center Z of the cap is always covered by any d -close cap. Thus, we see that any internal layer Sn−1(A ) of the cap C(μ, Z) has a smaller uncovered fraction Ω ≤ Ω . μ This gives the required condition p(x) ≤ Ω ≤ Ω < ω for any point x and proves our lemma. Remark Our choice of d in (22) is central to the above proof, and even a marginal increase in d will completely change our setting. Namely, it can be proven that about half the base of the μ-cap is uncovered if a ρ-cap is (d + ε)-close. Our next goal is to estimate the expected number N of non-saturated caps C(μ, z ) left after N trials. Lemma 7 For n ≥ 100, the expected number N of non-saturated caps C(μ, z ) is N < 2−n/4N . (31) Proof Given any center z, a randomly chosen center y is d -close to z with the probability Ωd . Then the probability to obtain s or fewer such caps is P = s i=0 N i Ωdi (1 − Ωd )N−i . Note that for any i ≤ s, (1 − Ωd )N−i ≤ exp{−(Ωd + Ωd2/2)(N − i)}. Now we use inequality Ωd ≤ 1/2 and the lower bound N Ωd > λn ln n − Ωd of (25). Then (Ωd + Ωd2/2)(N − i) Ωd = N Ωd 1 + 2 Ωd − iΩd 1 + 2 ≥ (λn ln n − Ωd ) + Ω2d [λn ln n − Ωd − s(2 + Ωd )] ≥ λn ln n. Here last inequality follows from the fact that s(2 + Ωd ) ≤ 5n/(6 ln ln n) and therefore Then λn ln n − Ωd − s(2 + Ωd ) ≥ 2. P ≤ e−λn ln n s (Ωd N )i i=0 i! ≤ e−λn ln n . i=0 i! (32) Note that consecutive summands in (32) differ at least (λn ln n)/s ≥ λq ln n times. Therefore ≤ (eλq ln n)n/q . Here the sum of the geometric series (λq ln n)−i was first bounded from above by cn < c100 < 2. Then we used the Sterling formula in the form s! > (2π s)1/2(s/e)s and removed (for simplicity) the vanishing term 2(2π s)−1/2. Finally, the last inequality follows from the fact that its left-hand side increases in s for any s < λn ln n. Summarizing, these substitutions give where P ≤ exp{nhn − λn ln n}, hn = Next, we recalculate ϑ∗ of (16) using parameter λ of (22) and the lower bound of (25). It is easy to verify that for any n, Now we estimate the size of Cov(μ) using ( 8 ) and (22) as follows |Cov(μ)| ≤ ϑ∗/Ωμ ≤ 2N Ωd /Ωμ ≤ 2N μn ≤ 2N exp{n[β ln n + ln(12)/2]}. (33) Thus, the expected number of non-saturated caps is N ≤ |Cov(μ)|P ≤ 2N exp{n[hn − (λ − β) ln n + ln(12)/2]} ≤ 2N exp{n[hn − (5 − ln 12)/2]}. (34) Now we see that the quantity Ψn in the brackets of (34) consists of the declining positive function hn and the negative constant. Thus, Ψn is a declining function of n. Direct calculation shows that Ψ100 < −0.257. Therefore estimate (31) holds. Consider now the saturated caps C(μ, z) and the centers u ∈ Cov(ε) inside them. Lemma 8 For any n ≥ 100, the number of centers u ∈ Cov(ε) left uncovered in all saturated caps C(μ, z) has expectation N < 2−n/2N . Proof We first estimate the total number |Cov(ε)| of centers u. Similarly to (33), |Cov(ε)| ≤ ϑ∗/Ωε ≤ 2N Ωd /Ωε ≤ 2N (2n ln n)n. Any cap C(μ, z) intersects with at least s + 1 randomly chosen caps C(ρ , y). According to Lemma 6, any single ρ-cap fails to cover any given point x ∈ C(μ, z) with probability ω or less. Therefore any point u ∈ C(μ, z) is not covered with probability ωs+1 or less. Note that ωs+1 < ωn/q where q = 3 ln ln n. Then we use the upper bound (27) for ω and deduce that where N ≤ |Cov(ε)| · ωn/q ≤ 2N exp{−nΦn}, Φn = ln2 n ln 4 2 ln ln n + 3 ln ln n 1 − ln n − ln ln n − ln 2 + 3 . (35) (36) Direct calculation shows that Φ100 > 0.71. It is also easy to see that the first term of Φn (in parentheses) moderates both terms ln n and ln ln n. Thus, Φn > 0.71 for all n ≥ 100, and the lemma is proved. Proof of Theorem 1 Consider any cap C(μ, z¯) that contains at least one uncovered center u ∈Cov(ε). Such a cap is either non-saturated or porous and therefore {¯} = {z } ∪ {z }. (Equivalently, we can directly consider the set {z } ∪ {u }.) Then, z according to Lemmas 7 and 8, {z¯} has expected size N¯ ≤ N + N < 21−n/4N . Thus, there exist N randomly chosen centers y that leave at most 21−n/4N centers z¯. The extended set {x} = {y} ∪ {z¯} forms a unit covering of Srn. This covering has density ϑ ≤ Ω1(N + N¯ ) ≤ Ω1N (1 + 21−n/4) ≤ λn ln n(1 + 21−n/4)Ω1/Ωd . Similarly to inequality (9), we now verify that for n ≥ 100, Ω1/Ωd ≤ 1 1 − n ln n − n−2β −n Indeed, replace parameter t = ln1n used in the proof of Lemma 4 with the new value 1 1 1 t = ln n + n1−2β = ln n + ln4 n . This new t also satisfies inequality (11) for n ≥ 100, which in turn proves inequality (37). As a secondary remark, note that (37) also proves asymptotic equality (24). Finally, we take λ of (9) and combine the last inequalities for ϑ and Ω1/Ωd as follows 5 + 2 ln n 1 1 1 + ln n + ln2 n (1 + 21−n/4) 5 + ln n . Here we again used Lemma 3. Namely, we verify numerically that the last expression exceeds the previous one at n = 100 and moderates it for larger n, due to its bigger remaining term 5/ ln n. This completes the non-asymptotic case n ≥ 3. To complete the proof of Theorem 1, we now present similar estimates for n → ∞. We take any constant b > 3/2 and redefine the parameters in (22) and (26) as follows: 1 ln ln n β = 2 + b ln n 3 1 λ = β + 4 ln n , , q = ln2 ln n. First, bounds (30) and (27) can be replaced with Second, bound (34) can be rewritten as Note that the first term ln(eλq ln n)/q vanishes for n → ∞, and N declines exponentially in n. Thus, Lemma 7 holds. Finally, bound (35) is replaced with N ≤ 2N exp n . , which vanishes (faster than exponent in n) for any given b > 3/2. Thus, Lemma 8 also holds. Then we proceed similarly to (37). In this case, for sufficiently large n, we obtain the density which proves ( 6 ). This completes the proof of Theorem 1. For finite n, we can slightly refine bounds (28–29), and also use exact expression for ω in (27). However, these refinements only marginally improve the main bound ( 5 ), which is replaced with ϑ 1 n ln n ≤ 2 + + 4 ln ln n ln n . Finally, note that Theorem 1 directly leads to Corollary 2. Indeed, here we can use the well known fact ϑ (Rn−1) = rl→im∞ ϑ (Srn) (see a sketch of the proof in [6] or Theorem detailed for packings of Rn). .1 in [7], where a similar proof is Concluding remarks In summary, we prove in this paper that the classic Rogers bound ( 3 ) on covering density of a sphere Srn or the Euclidean space Rn can be reduced about two times for large dimensions n. The main open problem is to further reduce the gap between this bound and its lower counterpart ( 2 ), which is linear in n. In this regard, note that our design holds if we employ any constant parameter β > 1/2 introduced in (22). However, it can be verified that choosing a smaller constant β < 1/2 will first increase parameter μ = n−β and then reduce parameter d of (22) to the order of 1 − μ2/2. The latter reduction will exponentially increase the covering size, by a factor of Ωρ /Ωd → exp{1 − 2β}. Therefore, our conjecture is that a completely new design is needed for further asymptotic reductions. Another important problem is to extend the above results to the balls Brn of an arbitrary radius r . Our conjecture is that ϑ (Brn) ≤ ( 12 + o( 1 ))n ln n for any r and Acknowledgements The author is grateful to an anonymous referee for many helpful comments and remarks. This research was supported in part by NSF grants CCF-0622242 and CCF-0635339. 1. Coxeter , H.S.M. , Few , L. , Rogers , C.A. : Covering space with equal spheres . Mathematika 6 , 147 - 157 ( 1959 ) 2. Rogers , C.A. : A note on coverings . Mathematika 4 , 1 - 6 ( 1957 ) 3. Rogers , C.A. : Covering a sphere with spheres . Mathematika 10 , 157 - 164 ( 1963 ) 4. Böröczky , K. Jr. , Wintsche , G.: Covering the sphere with equal spherical balls . In: Aronov, B. , Bazú , S. , Sharir , M. , Pach , J . (eds.) Discrete Computational Geometry-The Goldman-Pollack Festschrift , pp. 237 - 253 . Springer, Berlin ( 2003 ) 5. Verger-Gaugry , J.-L.: Covering a ball with smaller equal balls in Rn . Discrete Comput. Geom . 33 , 143 - 155 ( 2005 ) 6. Conway , J.H. , Sloane , N.J.A. : Sphere Packings, Lattices and Groups . Springer, New York ( 1988 ) 7. Levenstein , V.I.: Bounds on packings of metric spaces and some of their applications . Vopr. Kibern . 40 , 43 - 109 ( 1983 ) (in Russian) 8. Dumer , I. : Covering spheres and balls with smaller balls . In: 2006 IEEE International Symposium Information Theory , Seattle, WA, July 9- 15 2006 , pp. 992 - 995 ( 2006 )

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Ilya Dumer. Covering Spheres with Spheres, Discrete & Computational Geometry, 2007, 665-679, DOI: 10.1007/s00454-007-9000-7