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


Authors: Lauren Huckaba, Neil Lyall and Ákos Magyar
Journal: Proc. Amer. Math. Soc. 145 (2017), 2335-2347
MSC (2010): Primary 11B30, 42B25, 42A38
DOI: https://doi.org/10.1090/proc/13538
Published electronically: January 25, 2017
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Abstract: 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.


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Additional Information

Lauren Huckaba
Affiliation: Department of Mathematics, The University of Georgia, Athens, Georgia 30602
Email: lhuckaba@math.uga.edu

Neil Lyall
Affiliation: Department of Mathematics, The University of Georgia, Athens, Georgia 30602
Email: lyall@math.uga.edu

Ákos Magyar
Affiliation: Department of Mathematics, The University of Georgia, Athens, Georgia 30602
Email: magyar@math.uga.edu

DOI: https://doi.org/10.1090/proc/13538
Received by editor(s): April 1, 2016
Received by editor(s) in revised form: July 21, 2016
Published electronically: January 25, 2017
Communicated by: Alexander Iosevich
Article copyright: © Copyright 2017 American Mathematical Society