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Degree reduction and graininess for Kakeya-type sets in \(\mathbb{R}^3\). (English) Zbl 1358.52011

In [Adv. Math. 225, No. 5, 2828–2839 (2010; Zbl 1202.52022)], the author and N. H. Katz proved some results regarding the combinatorics of a finite set of lines in the 3-dimensional space \({\mathbb R}^3\), by using the polynomial method. In the paper under review, the author is able to adapt some ideas contained in the above mentioned work in order to show results for cylindrical tubes instead of lines in \({\mathbb R}^3\). Thus he proves that if \({\mathfrak T}\) is a set of tubes in \({\mathbb R}^3\) with length \(N\) and radius 1 satisfying that (i) the union of the cylinders has volume \(N^{3-\sigma}\), (ii) most points in the union are contained in three tubes of \({\mathfrak T}\) pointing in quantitatively different directions, then in a typical ball of radius \(N^{\sigma}\) the union of the tubes resembles a collection of rectangular slabs of dimensions \(1\times N^{\sigma}\times N^{\sigma}\).

MSC:

52A35 Helly-type theorems and geometric transversal theory
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
52A37 Other problems of combinatorial convexity

Citations:

Zbl 1202.52022
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Full Text: DOI arXiv

References:

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[9] Stone, A. H. and Tukey, J. W.: Generalized ”sandwich” theorems. Duke Math. J. 9 (1942), 356–359. Received May 13, 2014. Larry Guth: Massachusetts Institute of Technology, Department of Mathematics, 77 Massachusetts Ave, Cambridge MA 02139, USA. E-mail:larry.guth.work@gmail.com IntroductionPlaniness and graininessStatement of resultsDegree reductionSimple examplesMain ideas of the proofOrganization of the paperParameter counting and the vanishing lemma for tubesDegree reduction for tubesDegree reduction for linesDegree reduction for tubesBackground in integral geometryPlaniness and graininess estimatesReasonable cubesThe curvature of Z on reasonable tube segmentsSlices of reasonable tube segmentsCurvature estimates in non-straight directionsPointwise curvature boundsThe end of the proof
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