Simplices and sets of positive upper density in \mathbb{R}^d

We prove an extension of Bourgain's theorem on pinned distances in a measurable subset of \mathbb{R}^2 of positive upper density, namely Theorem 1^\prime in a 1986 article, to pinned non-degenerate k-dimensional simplices in a measurable subset of \mathbb{R}^{d} of positive upper density whenev...

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Bibliographic Details
Published in:Proceedings of the American Mathematical Society 2017-06, Vol.145 (6), p.2335-2347
Main Authors: Huckaba, Lauren, Lyall, Neil, Magyar, Ákos
Format: Article
Language:English
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Summary:We prove an extension of Bourgain's theorem on pinned distances in a measurable subset of \mathbb{R}^2 of positive upper density, namely Theorem 1^\prime in a 1986 article, to pinned non-degenerate k-dimensional simplices in a measurable subset of \mathbb{R}^{d} of positive upper density whenever d\geq k+2 and k is any positive integer.
ISSN:0002-9939
1088-6826
DOI:10.1090/proc/13538