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

Abey López-García, Ryan E. McCleary

Published 2021-05-22Version 1

In this paper 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 on the circle at the point $a_{N}$. In this work, we analyze the asymptotic properties of these extremal values $U_{N,s}(a_N)$, treating separately the cases $0<s<1$, $s=1$, and $s>1$. We present new second-order asymptotic formulas for $U_{N,s}(a_N)$ in the cases $0<s<1$ and $s=1$. A first-order result for $s>1$ is proved, and it is shown that the first-order normalized sequence $(U_{N,s}(a_N)/N^s)_{N=1}^{\infty}$ is divergent in this case.