Polynomial Szemer\'edi for sets with large Hausdorff dimension on the Torus

TL;DR

This paper proves that large Hausdorff dimension sets on the torus contain polynomial configurations, extending combinatorial results to fractal sets, and also investigates convergence properties of polynomial ergodic averages.

Contribution

It establishes a polynomial Szemerédi type result for sets with large Hausdorff dimension on the torus, using Sobolev smoothing inequalities and ergodic theory techniques.

Findings

Sets with Hausdorff dimension greater than 1 - epsilon contain polynomial configurations.

Divergence set of polynomial ergodic averages has Hausdorff dimension less than one.

Method combines harmonic analysis, ergodic theory, and fractal geometry.

Abstract

Let $\mathbb{P}= \{P_1, \cdots, P_{k}\in \mathbb{R}[y]\}$ be a collection of polynomials with distinct degrees and zero constant terms. We proved that there exists $\epsilon=\epsilon(\mathbb{P})>0$ such that, for any compact set $E \subset \mathbb{T}$ with dim(E)$>1-\epsilon$, we can find $y\neq 0$ so that $\{x,x+P_1(y), \cdots,x+P_k(y)\} \subset E$. The proof relies on a suitable version of the Sobolev smoothing inequality with ideas adapted from Peluse \cite{P19}, Durcik and Roos \cite{DR24}, and Krause, Mirek, Peluse, and Wright \cite{KMPW24}. As a byproduct of our Sobolev smoothing inequality, we demonstrated that the divergence set of the pointwise convergence problem for certain polynomial multiple ergodic averages has Hausdorff dimension strictly less than one.