Asymptotics of the optimal values of potentials generated by greedy energy sequences on the unit circle
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| Format: | Preprint |
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2023
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| author | López-García, Abey Miña-Díaz, Erwin |
| author_facet | López-García, Abey Miña-Díaz, Erwin |
| contents | 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 $(F_n)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2309_04387 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Asymptotics of the optimal values of potentials generated by greedy energy sequences on the unit circle López-García, Abey Miña-Díaz, Erwin Classical Analysis and ODEs 31C20, 31A15 (Primary) 11M06 (Secondary) 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 $(F_n)$. |
| title | Asymptotics of the optimal values of potentials generated by greedy energy sequences on the unit circle |
| topic | Classical Analysis and ODEs 31C20, 31A15 (Primary) 11M06 (Secondary) |
| url | https://arxiv.org/abs/2309.04387 |