Asymptotics of the energy of sections of greedy energy sequences on the unit circle, and some conjectures for general sequences

TL;DR

This paper analyzes the asymptotic behavior of the Riesz s-energy of greedy energy sequences on the unit circle, providing first and second-order results and proposing conjectures for more general sequences.

Contribution

It offers new asymptotic formulas for the energy of greedy sequences on the circle and introduces conjectures for broader classes of sequences.

Findings

First-order asymptotics of energy sequences

Second-order asymptotic results

Energy expressed via binary representation of N

Abstract

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.