An improved bound on the Hausdorff dimension of Besicovitch sets in ℝ³

By Nets Hawk Katz, Joshua Zahl. Journal of The American Mathematical Society, 2017

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<p>We prove that every Besicovitch set in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R cubed"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb {R}^3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> must have Hausdorff dimension at least <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="5 slash 2 plus epsilon 0"> <mml:semantics> <mml:mrow> <mml:mn>5</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>ϵ</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">5/2+\epsilon _0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for some small constant <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="epsilon 0 greater-than 0"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>ϵ</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\epsilon _0>0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. This follows from a more general result about the volume of unions of tubes that satisfies the Wolff axioms. Our proof grapples with a new “almost counterexample” to the Kakeya conjecture, which we call the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S upper L 2"> <mml:semantics> <mml:msub> <mml:mi>SL</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">\operatorname {SL}_2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> example; this object resembles a Besicovitch set that has Minkowski dimension 3 but Hausdorff dimension <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="5 slash 2"> <mml:semantics> <mml:mrow> <mml:mn>5</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">5/2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We believe this example may be an interesting object for future study.</p>

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/40bf4caa754bf951efd79865d0c2f12368160355