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

TL;DR

This paper analyzes the asymptotic behavior of the minimal potential values generated by greedy energy sequences on the unit circle for Riesz and logarithmic potentials, revealing detailed limit properties and the structure of their limit points.

Contribution

It provides exact asymptotic formulas for the minimal potential values and characterizes the set of all their limit points, advancing understanding of greedy energy sequences on the circle.

Findings

Determined the asymptotic behavior of potential minima $U_n$ as $n o

Established the exact value of $\

,

Abstract

For the Riesz and logarithmic potentials, we consider greedy energy sequences $(a_n)_{n=0}^\infty$ on the unit circle $S^1$, constructed in such a way that for every $n\geq 1$, the discrete potential generated by the first $n$ points $a_0,\ldots,a_{n-1}$ of the sequence attains its minimum value (say $U_n$) at $a_n$. We obtain asymptotic formulae that describe the behavior of $U_n$ as $n\to\infty$, in terms of certain bounded arithmetic functions with a doubling periodicity property. As previously shown in \cite{LopMc2}, after properly translating and scaling $U_n$, one obtains a new sequence $(F_n)$ that is bounded and divergent. We find the exact value of $\liminf F_n$ (the value of $\limsup F_n$ was already given in \cite{LopMc2}), and show that the interval $[\liminf F_n,\limsup F_n]$ comprises all the limit points of the sequence $(F_n)$.