Year: 2025
Type: preprint
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 $ε=ε(\mathbb{P})>0$ such that, for any compact set $E \subset \mathbb{T}$ with dim(E)$>1-ε$, 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.
Source: arXiv (Cornell University)
Author: Hong, Guo-Dong
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4417429120
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