Higher-order asymptotics for the energy of greedy sequences on the unit circle

TL;DR

This paper derives higher-order asymptotic expansions for the energy of greedy point sequences on the unit circle, revealing oscillatory behavior and the distribution of their limit points.

Contribution

It provides new asymptotic formulas for the energy of greedy sequences, including a simpler proof of the density of their potential values.

Findings

Asymptotic expansion involves oscillatory sequences with doubling periodicity.

The scaled energy sequence diverges but has limit points filling an interval.

Density results for potential values of greedy sequences are established.

Abstract

For the Riesz and logarithmic energies, we consider a greedy sequence $(a_n)_{n=0}^\infty$ of points on the unit circle $S^1$ constructed in such a way that for every integer $N\geq 2$, the energy of the configuration $(a_0,\ldots,a_{N-2},x)$ attains its optimal value (say $E_N$) at $x=a_{N-1}$. We derive an asymptotic expansion for $E_N$ in terms of certain bounded, oscillatory sequences $H_{N}$, $K_{N}$, and $R_{N}$ with a doubling periodicity property. In particular, we recover the results of \cite{LopMc1,LopWag} showing that after a proper translation and scaling of $E_N$, one is left with a sequence $T_N$ that is bounded and divergent. We show that the limit points of the sequence $T_N$ fill a closed interval. This follows from our asymptotic formulae and an analogous density result for the limit points of the sequences $H_{N}$, $K_{N}$, and $R_{N}$. We also give a new, simpler proof of density results obtained in \cite{LopMin} for the optimal values of the potential generated by a greedy sequence.