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arXiv · 1509.09283

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

Lauren Huckaba·Neil Lyall·Akos Magyar
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Abstract

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

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Lauren Huckaba, Neil Lyall, Akos Magyar. 2015-09-30. Simplices and sets of positive upper density in $\mathbb{R}^d$. https://arxiv.org/abs/1509.09283

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