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arXiv · 2309.04387

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

Abey López-García·Erwin Miña-Díaz
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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 $(F_n)$.

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Abey López-García, Erwin Miña-Díaz. 2023-09-08. Asymptotics of the optimal values of potentials generated by greedy energy sequences on the unit circle. https://arxiv.org/abs/2309.04387

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