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<p>Katz and Zahl [<CitationRef CitationID="CR18">18</CitationRef>] used a planebrush argument to prove that Kakeya sets in <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\mathbb {R}^4\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>4</mn> </msup> </math></EquationSource> </InlineEquation> have Hausdorff dimension at least 3.059. In the special case when the Kakeya set is <i>plany</i>, their argument gives a better lower bound of 10/3. We give a nontechnical exposition of the Katz-Zahl argument for plany Kakeya sets in the finite field setting.</p>

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A bound for plany Kakeya sets in \(\mathbb {F}_q^4\) using the planebrush method

  • 期刊论文

  • 发表日期：

    2026年1月10日

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  • Izabella Łaba,
  • Mukul Rai Choudhuri,
  • Joshua Zahl

摘要

Katz and Zahl [18] used a planebrush argument to prove that Kakeya sets in \(\mathbb {R}^4\) R 4 have Hausdorff dimension at least 3.059. In the special case when the Kakeya set is plany, their argument gives a better lower bound of 10/3. We give a nontechnical exposition of the Katz-Zahl argument for plany Kakeya sets in the finite field setting.