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<p>We describe several randomized collections of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(3\times 3\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>3</mn> <mo>×</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation> rotation matrices and analyze their associated logarithmic energy. The best one (that is, the one attaining the lowest expected logarithmic energy) is constructed by choosing <i>r</i> points on the sphere, which come from the zeros of a randomly chosen degree <i>r</i> polynomial, and considering at each of these points a set of <i>s</i> evenly distributed rotation matrices. This construction yields a new upper bound on the minimal logarithmic energy of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(n=rs\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>=</mo> <mi>r</mi> <mi>s</mi> </mrow> </math></EquationSource> </InlineEquation> rotation matrices.</p>

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Points on \(\operatorname {SO}(3)\) with low logarithmic energy

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年4月2日

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  • Carlos Beltrán,
  • Federico Carrasco,
  • Damir Ferizović,
  • Pedro R. López-Gómez

摘要

We describe several randomized collections of \(3\times 3\) 3 × 3 rotation matrices and analyze their associated logarithmic energy. The best one (that is, the one attaining the lowest expected logarithmic energy) is constructed by choosing r points on the sphere, which come from the zeros of a randomly chosen degree r polynomial, and considering at each of these points a set of s evenly distributed rotation matrices. This construction yields a new upper bound on the minimal logarithmic energy of \(n=rs\) n = r s rotation matrices.