{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,23]],"date-time":"2026-10-23T05:58:23Z","timestamp":1792735103226,"version":"4.5.0"},"reference-count":19,"publisher":"American Mathematical Society (AMS)","issue":"1","license":[{"start":{"date-parts":[[2019,8,29]],"date-time":"2019-08-29T00:00:00Z","timestamp":1567036800000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"funder":[{"DOI":"10.13039\/501100000038","name":"Natural Sciences and Engineering Research Council of Canada","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100000038","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. Amer. Math. Soc."],"abstract":"<p>We prove that every Besicovitch set in <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"double-struck upper R cubed\">\n  <mml:semantics>\n    <mml:msup>\n      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n        <mml:mi mathvariant=\"double-struck\">R<\/mml:mi>\n      <\/mml:mrow>\n      <mml:mn>3<\/mml:mn>\n    <\/mml:msup>\n    <mml:annotation encoding=\"application\/x-tex\">\\mathbb {R}^3<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> must have Hausdorff dimension at least <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"5 slash 2 plus epsilon 0\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mn>5<\/mml:mn>\n      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n        <mml:mo>\/<\/mml:mo>\n      <\/mml:mrow>\n      <mml:mn>2<\/mml:mn>\n      <mml:mo>+<\/mml:mo>\n      <mml:msub>\n        <mml:mi>\u03f5<\/mml:mi>\n        <mml:mn>0<\/mml:mn>\n      <\/mml:msub>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">5\/2+\\epsilon _0<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> for some small constant <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"epsilon 0 greater-than 0\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:msub>\n        <mml:mi>\u03f5<\/mml:mi>\n        <mml:mn>0<\/mml:mn>\n      <\/mml:msub>\n      <mml:mo>&gt;<\/mml:mo>\n      <mml:mn>0<\/mml:mn>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">\\epsilon _0&gt;0<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/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 \u201calmost counterexample\u201d to the Kakeya conjecture, which we call the <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper S upper L 2\">\n  <mml:semantics>\n    <mml:msub>\n      <mml:mi>SL<\/mml:mi>\n      <mml:mn>2<\/mml:mn>\n    <\/mml:msub>\n    <mml:annotation encoding=\"application\/x-tex\">\\operatorname {SL}_2<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> example; this object resembles a Besicovitch set that has Minkowski dimension 3 but Hausdorff dimension <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"5 slash 2\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mn>5<\/mml:mn>\n      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n        <mml:mo>\/<\/mml:mo>\n      <\/mml:mrow>\n      <mml:mn>2<\/mml:mn>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">5\/2<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>. We believe this example may be an interesting object for future study.<\/p>","DOI":"10.1090\/jams\/907","type":"journal-article","created":{"date-parts":[[2018,7,25]],"date-time":"2018-07-25T14:26:38Z","timestamp":1532528798000},"page":"195-259","source":"Crossref","is-referenced-by-count":20,"title":["An improved bound on the Hausdorff dimension of Besicovitch sets in \u211d\u00b3"],"prefix":"10.1090","volume":"32","author":[{"given":"Nets","family":"Katz","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Joshua","family":"Zahl","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"14","published-online":{"date-parts":[[2018,8,29]]},"reference":[{"issue":"2","key":"1","doi-asserted-by":"publisher","first-page":"261","DOI":"10.1007\/s11511-006-0006-4","article-title":"On the multilinear restriction and Kakeya conjectures","volume":"196","author":"Bennett, Jonathan","year":"2006","journal-title":"Acta Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0001-5962","issn-type":"print"},{"key":"2","doi-asserted-by":"publisher","first-page":"193","DOI":"10.1007\/s11854-010-0028-x","article-title":"The discretized sum-product and projection theorems","volume":"112","author":"Bourgain, Jean","year":"2010","journal-title":"J. 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