Iterated differences sets, diophantine approximations and applications

TL;DR

The paper develops a Diophantine-approximation framework for odd real polynomials by studying the sets $\mathcal{R}(v,\varepsilon)=\{n\in\mathbb{N}: \|v(n)\|<\varepsilon\}$ and their largeness properties in the ultrafilter and combinatorial senses. It proves that, for an odd polynomial $v$ of degree $2\ell-1$, $\mathcal{R}(v,\varepsilon)$ is $\Delta_\ell^*$ (and often $\Delta_\ell$-type), with a sharp converse: if this holds then $\deg v\le 2\ell-1$ (under irrational leading coefficient). A central achievement is a full characterization: $\mathcal{R}(v,\varepsilon)$ is $\Delta_\ell^*$ precisely when $v-w$ is odd of degree at most $2\ell-1$ for some rational $w$ with $w(0)\in\mathbb{Z}$. The results yield new recurrence properties for polynomial iterates in ergodic theory, including weak-mixing characterizations and Furstenberg–Sárközy-type statements, and they establish the sharpness of several largeness notions through counterexamples and a detailed hierarchy of set families.

Abstract

Let $v$ be an odd real polynomial (i.e. a polynomial of the form $\sum_{j=1}^\ell a_jx^{2j-1}$). We utilize sets of iterated differences to establish new results about sets of the form $\mathcal R(v,ε)=\{n\in\mathbb{N}\,|\,\|v(n)\|{<ε\}}$ where $\|\cdot\|$ denotes the distance to the closest integer. We then apply the new diophantine results to obtain applications to ergodic theory and combinatorics. In particular, we obtain a new characterization of weakly mixing systems as well as a new variant of Furstenberg-Sárközy theorem.

Theorems & Definitions (88)

• Proposition 1.1
• Theorem 1.2
• Remark 1.3
• Theorem 1.4
• Theorem 1.5
• Corollary 1.6
• Theorem 1.7
• Remark 1.8
• Corollary 1.9
• Corollary 1.10