单位圆上贪婪能量序列截面能量的渐近性及对一般序列的几点证明
Asymptotics of the energy of sections of greedy energy sequences on the unit circle, and some conjectures for general sequences
摘要：
本文研究了单位圆上贪婪$s$ -能量序列的第一个$N$点的Riesz $s$ -能量的渐近行为，对于$s$在$0\leq s<\infty$范围内的所有值（通常识别具有对数能量的$s=0$的情况）.在单位圆的情况下，贪婪$s$ -能量序列与利用对数势构造的经典Leja序列一致.我们得到了一阶和二阶渐近结果。其关键思想是用$N$的二进制表示来表示贪婪$s$ -能量序列的第一个$N$点的Riesz $s$ -能量。受这些结果的启发，我们给出了单位圆上一般序列的一些构造。
英文摘要：
In this paper we investigate the asymptotic behavior of the Riesz $s$-energy of the first $N$ points of a greedy $s$-energy sequence on the unit circle, for all values of $s$ in the range $0\leq s<\infty$ (identifying as usual the case $s=0$ with the logarithmic energy). In the context of the unit circle, greedy $s$-energy sequences coincide with the classical Leja sequences constructed using the logarithmic potential. We obtain first-order and second-order asymptotic results. The key idea is to express the Riesz $s$-energy of the first $N$ points of a greedy $s$-energy sequence in terms of the binary representation of $N$. Motivated by our results, we pose some conjectures for general sequences on the unit circle.
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