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# Asymptotics for Minimal Discrete Riesz Energy on Curves in $\R^d$

Published:2004-06-01
Printed: Jun 2004
• A. Martínez-Finkelshtein
• V. Maymeskul
• E. A. Rakhmanov
• E. B. Saff
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## Abstract

We consider the $s$-energy $$E(\ZZ_n;s)=\sum_{i \neq j} K(\|z_{i,n}-z_{j,n}\|;s)$$ for point sets $\ZZ_n=\{ z_{k,n}:k=0,\dots,n\}$ on certain compact sets $\Ga$ in $\R^d$ having finite one-dimensional Hausdorff measure, where $$K(t;s)= \begin{cases} t^{-s} ,& \mbox{if } s>0, \\ -\ln t, & \mbox{if } s=0, \end{cases}$$ is the Riesz kernel. Asymptotics for the minimum $s$-energy and the distribution of minimizing sequences of points is studied. In particular, we prove that, for $s\geq 1$, the minimizing nodes for a rectifiable Jordan curve $\Ga$ distribute asymptotically uniformly with respect to arclength as $n\to\infty$.
 Keywords: Riesz energy, Minimal discrete energy, Rectifiable curves, Best-packing on curves
 MSC Classifications: 52A40 - Inequalities and extremum problems 31C20 - Discrete potential theory and numerical methods

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