29,291 research outputs found

    Counting joints in vector spaces over arbitrary fields

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    We give a proof of the "folklore" theorem that the Kaplan--Sharir--Shustin/Quilodr\'an result on counting joints associated to a family of lines holds in vector spaces over arbitrary fields, not just the reals. We also discuss a distributional estimate on the multiplicities of the joints in the case that the family of lines is sufficiently generic.Comment: Not intended for publication. References added and other minor edits in this versio

    On the Dimension of Finite Point Sets I. An Improved Incidence Bound for Proper 3D sets

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    We improve the well-known Szemer\'edi-Trotter incidence bound for proper 3--dimensional point sets (defined appropriately)Comment: Unpublished manuscript from the late Gyuri Elekes. It was edited on 2007.04.27. last tim

    On the Multilinear Restriction and Kakeya conjectures

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    We prove dd-linear analogues of the classical restriction and Kakeya conjectures in Rd\R^d. Our approach involves obtaining monotonicity formulae pertaining to a certain evolution of families of gaussians, closely related to heat flow. We conclude by giving some applications to the corresponding variable-coefficient problems and the so-called "joints" problem, as well as presenting some nn-linear analogues for n<dn<d.Comment: 38 pages, no figures, submitte
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