A note on configurations in sets of positive density which occur at all large scales

TL;DR

This paper introduces a dynamical systems approach to understanding configurations in large-scale dense sets, providing new conditions for the occurrence of geometric patterns and extending previous results in Euclidean combinatorics.

Contribution

It offers an alternative ergodic-theoretic framework using weak-* topology on function spaces, leading to a new criterion for configurations in positive density sets and a novel proof for specific geometric configurations.

Findings

Almost every function is asymptotically constant at large scales under translation-invariant measures.

New sufficient condition for configurations to occur in large-scale dense sets.

Application to three-point configurations with specified area and side length.

Abstract

Furstenberg, Katznelson and Weiss proved in the early 1980s that every measurable subset of the plane with positive density at infinity has the property that all sufficiently large real numbers are realised as the Euclidean distance between points in that set. Their proof used ergodic theory to study translations on a space of Lipschitz functions corresponding to closed subsets of the plane, combined with a measure-theoretical argument. We consider an alternative dynamical approach in which the phase space is given by the set of measurable functions from $\mathbb{R}^d$ to $[0,1]$, which we view as a compact subspace of $L^\infty(\mathbb{R}^d)$ in the weak-* topology. The pointwise ergodic theorem for $\mathbb{R}^d$-actions implies that with respect to any translation-invariant measure on this space, almost every function is asymptotically close to a constant function at large scales. This observation leads to a general sufficient condition for a configuration to occur in every set of positive upper Banach density at all sufficiently large scales, extending a recent theorem of B. Bukh. To illustrate the use of this criterion we apply it to prove a new result concerning three-point configurations in measurable subsets of the plane which form the vertices of a triangle with specified area and side length, yielding a new proof of a result related to work of R. Graham.