Asymptotics of the minimum values of Riesz and logarithmic potentials generated by greedy energy sequences on the unit circle

TL;DR

This paper studies the asymptotic behavior of the minimum potential values generated by greedy energy sequences on the unit circle for logarithmic and Riesz potentials, providing detailed second-order formulas for various parameter ranges.

Contribution

It offers new second-order asymptotic formulas for the extremal potential values for different s, extending known first-order results and analyzing generalized greedy sequences.

Findings

Second-order asymptotics for s=0, 0<s<1, s=1

Boundedness and divergence of normalized potential for s>1

First-order asymptotics for generalized greedy sequences with p≥1

Abstract

In this work we investigate greedy energy sequences on the unit circle for the logarithmic and Riesz potentials. By definition, if $(a_n)_{n=0}^{\infty}$ is a greedy $s$-energy sequence on the unit circle, the Riesz potential $U_{N,s}(x):=\sum_{k=0}^{N-1}|a_k-x|^{-s}$, $s>0$, generated by the first $N$ points of the sequence attains its minimum value at the point $a_{N}$, for every $N\geq 1$. In the case $s=0$ we minimize instead the logarithmic potential $U_{N,0}(x):=-\sum_{k=0}^{N-1}\log |a_{k}-x|$. We analyze the asymptotic properties of these extremal values $U_{N,s}(a_N)$, studying separately the cases $s=0$, $0<s<1$, $s=1$, and $s>1$. We obtain second-order asymptotic formulas for $U_{N,s}(a_N)$ in the cases $s=0$, $0<s<1$, and $s=1$ (the corresponding first-order formulas are well known). A first-order result for $s>1$ is proved, and it is shown that the normalized sequence $U_{N,s}(a_N)/N^s$ is bounded and divergent in this case. We also consider, briefly, greedy energy sequences in which the minimization condition is required starting from the point $a_{p+1}$ (instead of the point $a_{1}$ as previously stated), for some $p\geq 1$. For this more general class of greedy sequences, we prove a first-order asymptotic result for $0\leq s<1$.