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<p>We prove that, for every polyhedral or <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(C^1\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation> norm on <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\mathbb {R}^d\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> </math></EquationSource> </InlineEquation> and every set <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(E \subseteq \mathbb {R}^d\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>E</mi> <mo>⊆</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> of packing dimension <i>s</i>, the packing dimension of the distance set of <i>E</i> with respect to that norm is at least <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(\tfrac{s}{d}\)</EquationSource> <EquationSource Format="MATHML"><math> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>s</mi> <mi>d</mi> </mfrac> </mstyle> </math></EquationSource> </InlineEquation>. One of the main tools is a nonlinear projection theorem extending a result of M.&#xa0;Järvenpää. An explicit construction follows, demonstrating that these distance sets bounds are sharp for a large class of polyhedral norms.</p>

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On the packing dimension of distance sets with respect to \({\varvec{C^{1\!}}}\) and polyhedral norms

  • 期刊论文

  • 发表日期：

    2025年11月21日

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  • Iqra Altaf,
  • Ryan Bushling,
  • Bobby Wilson

摘要

We prove that, for every polyhedral or \(C^1\) C 1 norm on \(\mathbb {R}^d\) R d and every set \(E \subseteq \mathbb {R}^d\) E ⊆ R d of packing dimension s, the packing dimension of the distance set of E with respect to that norm is at least \(\tfrac{s}{d}\) s d . One of the main tools is a nonlinear projection theorem extending a result of M. Järvenpää. An explicit construction follows, demonstrating that these distance sets bounds are sharp for a large class of polyhedral norms.