Quantitative twisted patterns in positive density subsets

TL;DR

This paper improves quantitative results on the structure of large difference sets under quadratic forms by using polynomial orbits and a new uniform Furstenberg-Sárközy theorem, avoiding deep non-quantitative group results.

Contribution

It introduces a novel approach using polynomial orbits and a uniform Furstenberg-Sárközy theorem to obtain quantitative structure results, replacing non-quantitative group-based methods.

Findings

Enhanced quantitative bounds on difference sets under quadratic forms.

Development of a uniform Furstenberg-Sárközy theorem for non-vanishing polynomials.

Application of density increment and exponential sum bounds for new results.

Abstract

We make quantitative improvements to recently obtained results on the structure of the image of a large difference set under certain quadratic forms and other homogeneous polynomials. Previous proofs used deep results of Benoist-Quint on random walks in certain subgroups of $\operatorname{SL}_r(\mathbb{Z})$ (the symmetry groups of these quadratic forms) that were not of a quantitative nature. Our new observation relies on noticing that rather than studying random walks, one can obtain more quantitative results by considering polynomial orbits of these group actions that are not contained in cosets of submodules of $\mathbb{Z}^r$ of small index. Our main new technical tool is a uniform Furstenberg-S\'{a}rk\"{o}zy theorem that holds for a large class of polynomials not necessarily vanishing at zero, which may be of independent interest and is derived from a density increment argument and Hua's bound on polynomial exponential sums.