Product of simplices and sets of positive upper density in ℝ^d

2016/05/16 by Neil Lyall, Lyall, Neil, Akos Magyar +2
Computer Science · Mathematics · #Advanced Banach Space Theory #Advanced Harmonic Analysis Research #Classical Analysis and ODEs (math.CA) #Combinatorics (math.CO) #FOS: Mathematics #Optimization and Variational Analysis #Point processes and geometric inequalities #math.CA #math.CO

paper · pdf · doi:10.48550/arxiv.1605.04890

Substantial revisions made. To appear in Math. Proc. Cambridge Philos. Soc

openalex publication_date 2016/05/16 · arxiv created 2017/01/22 · arxiv updated 2017/01/24 · openalex created_date 2022/08/30 · openalex updated_date 2026/07/28

Abstract

We establish that any subset of ℝ^d of positive upper Banach density necessarily contains an isometric copy of all sufficiently large dilates of any fixed two-dimensional rectangle provided d≥4. We further present an extension of this result to configurations that are the product of two non-degenerate simplices; specifically we show that if Δ_k1 and Δ_k2 are two fixed non-degenerate simplices of k₁+1 and k₂+1 points respectively, then any subset of ℝ^d of positive upper Banach density with d≥ k₁+k₂+6 will necessarily contain an isometric copy of all sufficiently large dilates of Δ_k1×Δ_k2. A new direct proof of the fact that any subset of ℝ^d of positive upper Banach density necessarily contains an isometric copy of all sufficiently large dilates of any fixed non-degenerate simplex of k+1 points provided d≥ k+1, a result originally due to Bourgain, is also presented.

Related

• Simplices and sets of positive upper density in ℝ^d
• A Note on John Simplex with Positive Dilation
• Counterexamples to generalizations of the Erdős B+B+t problem
• Sumsets of dense sets and sparse sets
• On positive cubature rules on the simplex and isometric embeddings