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<p>The notion of weak tiling played a key role in the proof of Fuglede's spectral set conjecture for convex domains, due to the fact that every spectral set must weakly tile its complement. In this paper, we revisit the notion of weak tiling and establish some geometric properties of sets that weakly tile their complement. If <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(A \subset \mathbb{R}^d\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo>⊂</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> is a convex polytope, we give a direct and self-contained proof that <i>A</i> must be symmetric and have symmetric facets. If <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(A \subset \mathbb{R}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo>⊂</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation> is a finite union of intervals, we give a necessary condition on the lengths of the gaps between the intervals.</p>

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Geometric implications of weak tiling

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年11月12日

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  • M. N. Kolountzakis,
  • N. Lev,
  • M. Matolcsi

摘要

The notion of weak tiling played a key role in the proof of Fuglede's spectral set conjecture for convex domains, due to the fact that every spectral set must weakly tile its complement. In this paper, we revisit the notion of weak tiling and establish some geometric properties of sets that weakly tile their complement. If \(A \subset \mathbb{R}^d\) A ⊂ R d is a convex polytope, we give a direct and self-contained proof that A must be symmetric and have symmetric facets. If \(A \subset \mathbb{R}\) A ⊂ R is a finite union of intervals, we give a necessary condition on the lengths of the gaps between the intervals.