Weak hypergraph regularity and applications to geometric Ramsey theory

TL;DR

This paper proves that sets with positive density in Euclidean space and integer lattices necessarily contain large dilates of certain geometric configurations, using a novel weak hypergraph regularity approach.

Contribution

It introduces a weak hypergraph regularity lemma and counting lemma applicable to Euclidean spaces and lattices, enabling new results in geometric Ramsey theory.

Findings

Sets of positive density contain large dilates of simplices.

Euclidean and lattice settings both exhibit these geometric configurations.

The methods extend hypergraph regularity to continuous and discrete spaces.

Abstract

Let $\Delta=\Delta_1\times\ldots\times \Delta_d\subseteq\mathbb{R}^n$, where $\mathbb{R}^n=\mathbb{R}^{n_1}\times\cdots\times\mathbb{R}^{n_d}$ with each $\Delta_i\subseteq\mathbb{R}^{n_i}$ a non-degenerate simplex of $n_i$ points. We prove that any set $S\subseteq \mathbb{R}^n$, with $n=n_1+\cdots +n_d$ of positive upper Banach density necessarily contains an isometric copy of all sufficiently large dilates of the configuration $\Delta$. In particular any such set $S\subseteq \mathbb{R}^{2d}$ contains a $d$-dimensional cube of side length $\lambda$, for all $\lambda\geq \lambda_0(S)$. We also prove analogous results with the underlying space being the integer lattice. The proof is based on a weak hypergraph regularity lemma and an associated counting lemma developed in the context of Euclidean spaces and the integer lattice.