Near-optimal density theorems for large dilates of large point configurations

TL;DR

This paper establishes near-optimal density thresholds for large sets in Euclidean space to contain all large similar copies of any n-point configuration, resolving a longstanding problem.

Contribution

It provides asymptotically sharp bounds for density thresholds in Euclidean and p spaces, matching known upper bounds up to logarithmic factors.

Findings

Lower bound of 1 - O((log n)/n) for Euclidean space

Asymptotically sharp bound of 1 - 1/n + o(1/n) for p spaces with p = 2

Use of equidistribution and probabilistic thinning in proofs

Abstract

We study density thresholds that force a measurable set $E\subseteq\mathbb{R}^d$ to contain all sufficiently large similar copies of every $n$-point configuration. We prove a lower bound of the form $1-O((\log n)/n)$, which matches the known upper bound up to the logarithmic factor, thus essentially resolving a problem posed by Falconer, Yavicoli, and the first author of the present paper. We also study the same problem for embeddings of $n$-point configurations into $\mathbb{R}^d$ equipped with the $\ell^p$ norm, obtaining an asymptotically sharp bound $1-1/n+o(1/n)$, as soon as $p\in(1,\infty)\setminus\{2\}$. In the proof of the former estimate we use equidistribution of polynomial sequences modulo $1$ combined with probabilistic thinning. The proof of the latter estimate relies on the geometry of the $\ell^p$ spaces for $p\neq2$.