Simplices and sets of positive upper density in $\mathbb {R}^d$
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- by Lauren Huckaba, Neil Lyall and Ákos Magyar PDF
- Proc. Amer. Math. Soc. 145 (2017), 2335-2347 Request permission
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.References
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Additional Information
- Lauren Huckaba
- Affiliation: Department of Mathematics, The University of Georgia, Athens, Georgia 30602
- MR Author ID: 984929
- Email: lhuckaba@math.uga.edu
- Neil Lyall
- Affiliation: Department of Mathematics, The University of Georgia, Athens, Georgia 30602
- MR Author ID: 813614
- Email: lyall@math.uga.edu
- Ákos Magyar
- Affiliation: Department of Mathematics, The University of Georgia, Athens, Georgia 30602
- MR Author ID: 318009
- Email: magyar@math.uga.edu
- 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
- © Copyright 2017 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 145 (2017), 2335-2347
- MSC (2010): Primary 11B30, 42B25, 42A38
- DOI: https://doi.org/10.1090/proc/13538
- MathSciNet review: 3626493