Asymptotic Properties of Discrete Minimal $s,log^t$-Energy Constants and Configurations

Combining the ideas of Riesz $s$-energy and $\log$-energy, we introduce the so-called $s,\log^t$-energy. In this paper, we investigate the asymptotic behaviors for $N,t$ fixed and $s$ varying of minimal $N$-point $s,\log^t$-energy constants and configurations of an infinite compact metric space of diameter less than $1$. In particular, we study certain continuity and differentiability properties of minimal $N$-point $s,\log^t$-energy constants in the variable $s$ and we show that in the limits as $s\rightarrow \infty$ and as $s\rightarrow s_0>0,$ minimal $N$-point $s,\log^t$-energy configurations tend to an $N$-point best-packing configuration and a minimal $N$-point $s_0,\log^t$-energy configuration, respectively. Furthermore, the optimality of $N$ distinct equally spaced points on circles in $\mathbb{R}^2$ for some certain $s,\log^t$ energy problems was proved.


Introduction
The general setting of discrete minimal energy problem is the following. Let (A, d) be an infinite compact metric space and K : A×A → R∪{∞} be a lower semicontinuous kernel. For a fixed set of N points ω N ⊂ A, we define the K-energy of ω N as follows E K (ω N ) := x =y x,y∈ω N K(x, y).
The minimal N-point K-energy of the set A is defined by where #ω N stands for the cardinality of the set ω N . A minimal N-point K-energy configuration is a configuration ω K N of N points in A that minimizes such energy, namely It is known that ω K N always exists and in general ω K N may not be unique. Two important kernels in the theory on minimal energy are Riesz and logarithmic kernels. The (Riesz) s-kernel and log-kernel are defined by and we denote by ω log N := ω K log N and call this configuration a minimal N-point logenergy configuration.
Let us provide a short survey of these two energy problems.
The study of s-energy constants and configurations has a long history in physics, chemistry, and mathematics. Finding the arrangements of ω s N where the set A is the unit sphere S 2 in the Euclidean space R 3 has been an active area since the beginning of the 19th century. The problem is known as the generalized Thomson problem (see [1] and [2,Chapter 2.4]). Candidates for ω s N for several numbers of N are available (see, e.g., [3]). However, the solutions (with rigorous proofs) are obtainable for handful values of N (see, e.g., [4,5,6, Author1(year)]). For a general compact set A in the Euclidean space R m , the study of the distribution of minimal N-point s-energy configurations of A as N → ∞ can be founded in [Author2(year)] and [Author3(year)]. In [Author3(year)], it was shown that when s is any fixed number greater than the Hausdorff dimension of A, minimal N-point s-energy configurations of A are "good points" to represent the set A in the sense that they are asymptotically uniformly distributed over the set A (see the precise statement in [Author3(year), Theorems 2.1 and 2.2]).
The log-energy problem has been heavily studied when A is a subset of the Euclidean space R 2 (or C) because it has had a profound influence in approximation theory (see, e.g., [7,8,9,10,11]). For A ⊂ C, the points in ω log N are commonly known as Fekete points or Chebyshev points which can be used as interpolation points (see [12]). The log-energy problem received another special attention when Steven Smale posed Problem #7 in his book chapter under the title "Mathematical problems for the next century" [13]. The problem #7 asks for a construction of an algorithm which on input N ≥ 2 outputs a configuration ω N = {x 1 , . . . , x N } of distinct points on S 2 embedded in R 3 such that (where c is a constant independent of N and ω N ) and requires that its running time grows at most polynomially in N. This arose form complexity theory in his joint work with Shub in [14]. In order to answer this question, it is natural to understand the asymptotic expansion of E log (S 2 , N) in the variable N (see [15] for conjectures and the progress). The problem concerning the arrangements of ω log N on the unit sphere S 2 in R 3 is posed by Whyte [16] in 1952. The Whyte's problem is also attractive and intractable. We refer to [17] for a glimpse of this problem.
In [2], Borodachov, Hardin, and Saff investigated asymptotic properties of minimal N-point s-energy constants and configurations for fixed N and varying s. Because this will be our main interest in this paper, we will state these results below.
The first theorem [ In order to state such theorem, let us define a set for s ≥ 0 Theorem A. Let (A, d) be an infinite compact metric space and let N ≥ 2 be fixed. Then, We will see in Theorems B and C below that there are certain relations between minimal s-energy problems, as s → 0 + , and best-packing problem defined as follows. The N-point best-packing distance of the set A is defined .
Before we state more results, let us define a cluster configuration. Let s 0 ∈ [0, ∞] We say that  (c) any cluster configuration of ω s N as s → ∞ is a N-point best-packing configuration.
In this paper, we consider the following s, log t -kernel with corresponding s, log t -energy of ω N and minimal N-point s, log t -energy of the set A respectively. We set and call it a minimal N-point s, log t -energy configuration. Note that the kernel K s log t (x, y) is lower semicontinuous on A × A and this s, log t -energy can be viewed as a generalization of both s-energy and log-energy. The kernel in (5) was first appeared in the study of the differentiability of the function f (s) in [2, Theorem 2.7.3]. To the authors' knowledge, no study involving s, log t -energy constants and configurations appears in the literature.
The main goal of this paper is to prove analogues of Theorems A, B, and C for s, log t -energy constants and configurations. We would like to emphasize that we will limit our interest to the sets A with diam(A) < 1, where denotes the diameter of A. For the cases where diam(A) ≥ 1, the values of the kernel K s log t (x, y) can be 0 or negative and the analysis becomes laborious. Furthermore, we investigate the arrangement of ω s,log t N on circles in R 2 for certain values of s and t.
An outline of this paper is as follows. The main results in this paper are stated in Section 2. We keep all auxiliary lemmas in Section 3. The proofs of the main results are in Section 4.

Main Results
Asymptotic behavior of minimal N-point s, log t -energy constants and configurations as s → ∞ can be explained in the following theorem.
Furthermore, every cluster configuration of ω s,log t N as s → ∞ is an N-point bestpacking configuration on A.
For a fixed t ≥ 0, we define The continuity of g(s) is stated below.
Analysis of cluster configurations of ω s,log t N as s → s 0 > 0 is in the following theorem.
For s ≥ 0 and t ≥ 0, we set The differentiability properties of g(s) are in Theorems 2.4 and 2.5. and Theorem 2.5. Let N ≥ 2 and t ≥ 0 be fixed. Assume that (A, d) is an infinite compact metric space with diam(A) < 1. Then, Let d u be the 2-dimensional Euclidean metric of R 2 . For α > 0, we denote by the circle centered at 0 of radius α. We let L(x, y) be the geodesic distance between the points x and y on S 1 α ; that is, the length of the shorter arc of S 1 α connecting the points x and y.
The optimality of N distinct equally spaced points on S 1 α with the Euclidean metric d u or the geodesic distance L for the certain s, log t -energy problems is stated in Propositions 2.1-2.3.
α with the geodesic distance L if and only if ω N is a configuration of N distinct equally spaced points on S 1 α . Proposition 2.3. Let N ≥ 2, s ≥ 0, t ≥ 1, and 0 < α < 1/2. Then, ω N is a minimal N-point s, log t -energy configuration on S 1 α with the Euclidean metric d u if and only if ω N is a configuration of N distinct equally spaced points on S 1 α . Note that the conditions 0 < α < π −1 in Proposition 2.1 and 0 < α < 1/2 in Proposition 2.3 are needed to make sure that diam(S 1 α ) < 1 corresponding to the Euclidean metric d u and the geodesic distance L, respectively. for all x ∈ (0, 1).
Since log

It follows that
Let Γ be a rectifiable simple closed curve in R m , m ≥ 2, of length |Γ| with a chosen orientation. We recall that L(x, y) is the geodesic distance between the points x and y on Γ. With the help of the following lemma [   Then, Let ω * N be a cluster configuration of ω s,log t N as s → ∞. This implies that there is a sequence {s k } ∞ k=1 ⊂ R such that s k → ∞ and ω s k ,log t N → ω * N as k → ∞. Arguing as in (8) Taking k → ∞, we obtain This means that ω * N is also an N-point best-packing configuration on A. Proof of Theorem 2.2. First of all, we show that g(s) is continuous on (0, ∞). Let s > 0 and let ω s,log t N be a minimal N-point s, log t -energy configuration on A. Using Lemma 3.4, we obtain for any ω s,log t N , lim inf and lim sup where the second inequality in (9) follows from the arbitrariness of ω s,log t N and the last inequality in (9) follows from Lemma 3.3.
Let ω N be a fixed configuration of N distinct points of A. Note that 0 < δ(ω N ) < 1. For all r ∈ (s/2, s), we have That is, This implies that for all r ∈ (s/2, s), is a strictly increasing function on (0, 1), there exists a constant c 2 > 0 such that for all r ∈ (s/2, s), Therefore, E s log t+1 (ω r,log t N ) are bounded above where r ∈ (s/2, s). From this and (10), Let s ≥ 0. Using Lemma 3.4, we also obtain for any ω s,log t N , lim sup and lim inf where the second inequality in (12) follows from rom the arbitrariness of ω s,log t N and the last inequality in (13) follows from Lemma 3.3.
The inequalities (9), (11), (12), and (13) imply that for all s > 0, and for all s ≥ 0 The inequalities in (14) and (15) further imply that g(s) is continuous for all s > 0 and is right continuous at s = 0.
Proof of Theorem 2.3. Let s 0 > 0. In order to show Theorem 2.3, it suffices to show that any cluster configuration of ω s,log t N as s → s + 0 or as s → s − 0 is a minimal N-point s 0 , log t -energy configuration on A.
Let ω * N be a cluster configuration of ω s,log t N , as s → s + 0 . Then, there is a sequence . For any configuration of N distinct points ω N on A, notice that α s E s log t (ω N ) is an increasing function of s. Applying the continuity of g(s) := E s log t (A, N) at s 0 , we have This implies that E s 0 log t (ω * N ) = E s 0 log t (A, N). Hence, ω * N is a minimal N-point s 0 , log tenergy configuration on A.
Let ω * * N be a cluster configuration of ω s,log t N , as s → s − 0 . Then, there is a sequence {s k } ∞ k=1 ⊂ [0, s 0 ) such that s k → s 0 and ω s k ,log t N → ω * * N as k → ∞. Without loss of generality, we may assume that s 0 /2 < s k < s 0 for all k. For any configuration of N distinct points ω N of A, observe that δ(ω N ) s E s log t (ω N ) is a decreasing function of s. It follows from the continuity of the function g(s) that g(s) is bounded above by some number M > 1 for all s ∈ (s 0 /2, s 0 ). For every s 0 /2 < s k < s 0 , Then, Using Lemma 3.1, there is a constant c > 0 such that Using the continuity of g(s) := E s log t (A, N) at s 0 , we have N). Hence, ω * * N is a minimal N-point s 0 , log tenergy configuration on A.
Proof of Theorem 2.4. Firstly, we show (6). Let s ≥ 0 be fixed and {r k } ∞ k=1 ⊂ (s, ∞) be a sequence such that r k → s as k → ∞ and Since A N is compact, there exists a subsequence and ω * N is a minimal N-point s, log t -energy configuration by Theorem 2.3. By , (13), (16), and (17), we get lim inf Then, It is easy to check that from Lemma 3.3, the constant inf G s log t+1 (A, N) in (19) is finite. This verifies (6).
Next, we prove (7). Let s > 0 be fixed and {r k } ∞ k=1 ⊂ [0, s) be a sequence such that r k → s as k → ∞ and Because A N is compact, there exists a subsequence {s ℓ } ∞ ℓ=1 ⊂ {r k } ∞ k=1 such that lim ℓ→∞ ω s ℓ ,log t N = ω * * N and ω * * N is a minimal N-point s, log t -energy configuration by Theorem 2.3. Then, we get lim Using (9), (10), (20), and (21), we obtain lim inf Then, Next, we want to show that sup G s log t+1 (A, N) is finite. Let ω N be a fixed configuration of N distinct points on A and let ω s,log t N be any minimal N-point s, log t configurations. Then,  N). = ω * * N . Then, ω * * N is a minimal N-point s 0 , log t -energy configuration by Theorem 2.3. Using (7) and the similar argument used to show (11), we have N).
This is a direct consequence of (b) and (c).
Proof of Proposition 2.1. Let N ≥ 2, s ≥ 0, t ≥ 1, and 0 < α < π −1 . We prove this proposition using Lemma 3.5. The function k : (0, 1) :→ R in the lemma is By Lemma 3.2, k(x) is strictly decreasing on (0, 1). Since for all x ∈ (0, 1), k(x) is strictly convex on (0, 1). Hence, because the function k(x) satisfies all required properties in Lemma 3.5, all minimal N-point K-energy configurations on S 1 α are configurations of N distinct equally spaced points on S 1 α with respect to the arc length and vice versa.
Proof of Proposition 2.2. Let N ≥ 2, 0 < α < (eπ) −1 , and s, t satisfy s > 0, t ≥ 0 or s = 0, t > 0. We can use the same lines of reasoning as in the proof of Proposition 2.1 except the function k is considered on (0, 1/e) and for all x ∈ (0, 1/e), Hence, because the function k(x) satisfies all required properties in Lemma 3.5, Proposition 2.2 is proved.

Discussion and Conclusions
We introduce minimal N-point s, log t -energy constants and configurations of an infinite compact metric space (A, d). Such constants and configurations are generated using the kernel In this paper, we study the asymptotic properties of minimal N-point s, log t -energy constants and configurations of A with diam(A) < 1, and N ≥ 2 and t ≥ 0 are fixed. We show that the s, log t -energy and every cluster configuration of ω s,log t N as s → ∞ is an N-point best-packing configuration on A. Furthermore, we show in Theorem 2.3 that for any s 0 > 0, any cluster configuration of ω s,log t N , as s → s 0 , is a minimal N-point s 0 , log t -energy configuration on A. When diam(A) < 1, our theorems generalize Theorems A, B, and C. The natural question would be "Do Theorems 2.1-2.5 hold true for diam(A) ≥ 1?" Investigation on arrangements of ω s N on circles in R 2 is in Propositions 2.1-2.3. In these propositions, we show that for certain values of s and t, all minimal Npoint s, log t -energy configurations on S 1 α with diam(S 1 α ) < 1 (corresponding to the Euclidean and geodesic distances) are the configurations of N distinct equally spaced points. We would like to report that the lemma 3.5 does not allow us to say something when diam(S 1 α ) ≥ 1. It would be very interesting to develop a new tool to attack the case when diam(S 1 α ) ≥ 1.