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

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...

Full description

Saved in:
Bibliographic Details
Published in:arXiv.org 2017-01
Main Authors: Huckaba, Lauren, Lyall, Neil, Magyar, Akos
Format: Article
Language:English
Subjects:
Density
Theorems
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary: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.
ISSN:2331-8422