Simplices and sets of positive upper density in ℝ^d

2015/09/30 by Lauren Huckaba, Huckaba, Lauren, Neil Lyall +3
Mathematics · #Classical Analysis and ODEs (math.CA) #Combinatorics (math.CO) #FOS: Mathematics #math.CA #math.CO

paper · pdf · doi:10.48550/arxiv.1509.09283

Minor revision made. To appear in Proc. Amer. Math. Soc

arxiv created 2017/01/07 · arxiv updated 2017/01/10

Abstract

We prove an extension of Bourgain's theorem on pinned distances in measurable subset of ℝ² of positive upper density, namely Theorem 1^′ in [Bourgain, 1986], to pinned non-degenerate k-dimensional simplices in measurable subset of ℝ^d of positive upper density whenever d≥ k+2 and k is any positive integer.

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