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Using elementary methods and some asymptotic analysis, we reprove known and prove new bounds (that are surprisingly close to conjectured bounds) for the minimal logarithmic and Riesz s-energy of point sets on the unit sphere in the Euclidean space \(\mathbb {R}^{d+1}\) for general \(d\geq 2\) . The novel approach works in the continuous case \(-s < s < 0\) , the logarithmic case and, in particular, in the singular case \(0 < s \leq d\) .

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Logarithmic and Riesz Energy on the Sphere: Better Bounds via Elementary Methods

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  • 发表日期：

    2026年

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  • Johann S. Brauchart

摘要

Using elementary methods and some asymptotic analysis, we reprove known and prove new bounds (that are surprisingly close to conjectured bounds) for the minimal logarithmic and Riesz s-energy of point sets on the unit sphere in the Euclidean space \(\mathbb {R}^{d+1}\) for general \(d\geq 2\) . The novel approach works in the continuous case \(-s < s < 0\) , the logarithmic case and, in particular, in the singular case \(0 < s \leq d\) .