An improved bound on the Hausdorff dimension of Besicovitch sets in ℝ³
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Bibliographic record
Abstract
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.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.005 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.003 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.002 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it