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Sets in $ \mathbb{R}^d$ with slow-decaying density that avoid an unbounded collection of distances


Author: Alex Rice
Journal: Proc. Amer. Math. Soc. 148 (2020), 523-526
MSC (2010): Primary 11B05, 28A75
DOI: https://doi.org/10.1090/proc/14802
Published electronically: October 18, 2019
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Abstract: For any $ d\in \mathbb{N}$ and any function $ f:(0,\infty )\to [0,1]$ with $ f(R)\to 0$ as $ R\to \infty $, we construct a set $ A \subseteq \mathbb{R}^d$ and a sequence $ R_n \to \infty $ such that $ \Vert x-y\Vert \neq R_n$ for all $ x,y\in A$ and $ \mu (A\cap B_{R_n})\geq f(R_n)\mu (B_{R_n})$ for all $ n\in \mathbb{N}$, where $ B_R$ is the ball of radius $ R$ centered at the origin and $ \mu $ is a Lebesgue measure. This construction exhibits a form of sharpness for a result established independently by Furstenberg-Katznelson-Weiss, Bourgain, and Falconer-Marstrand, and it generalizes to any metric induced by a norm on  $ \mathbb{R}^d$.


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Additional Information

Alex Rice
Affiliation: Department of Mathematics, Millsaps College, Jackson, Mississippi 39210
Email: riceaj@millsaps.edu

DOI: https://doi.org/10.1090/proc/14802
Received by editor(s): June 5, 2019
Published electronically: October 18, 2019
Communicated by: Alexander Iosevich
Article copyright: © Copyright 2019 American Mathematical Society