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On the Glasner Property of Linear Maps with Prime Entries on Tori

Andrew Rajchert

TL;DR

This work extends the quantitative Glasner property for maps between tori to higher dimensions and to matrices whose entries are polynomials evaluated at primes. It presents two main results: Theorem 1, where entries are indexed by independent primes and the lower bound on $k(\epsilon)$ scales as $\epsilon^{-4Lmn-\delta}$ (with a sharper $\epsilon^{-2Ln-\delta}$ when $m=1$), and Theorem 2, where all entries share a single prime and the bound scales as $\epsilon^{-(2L+1)(m+1)n-\delta}$ (with $m=1$ giving $\epsilon^{-(2L+1)n-\delta}$). The proofs adapt harmonic-analytic techniques, Hua-type exponential-sum bounds, and equidistribution results for polynomials at primes, introducing multiplicative complexity to handle the single-prime case. Together, these results deepen our understanding of how input dimension and prime-indexed arithmetic structure influence the density properties of linear maps on tori, with potential implications for arithmetic dynamics and uniform distribution questions on $\mathbb{T}^d$.

Abstract

We study the quantitative Glanser property in the context of maps between tori of differing dimension instead of as a (semi-)group action. We also only consider matrices with entries being non-constant polynomials evaluated at primes, extending on the work of Velani and Nair, and Bulinski and Fish to a more general setting.

On the Glasner Property of Linear Maps with Prime Entries on Tori

TL;DR

This work extends the quantitative Glasner property for maps between tori to higher dimensions and to matrices whose entries are polynomials evaluated at primes. It presents two main results: Theorem 1, where entries are indexed by independent primes and the lower bound on scales as (with a sharper when ), and Theorem 2, where all entries share a single prime and the bound scales as (with giving ). The proofs adapt harmonic-analytic techniques, Hua-type exponential-sum bounds, and equidistribution results for polynomials at primes, introducing multiplicative complexity to handle the single-prime case. Together, these results deepen our understanding of how input dimension and prime-indexed arithmetic structure influence the density properties of linear maps on tori, with potential implications for arithmetic dynamics and uniform distribution questions on .

Abstract

We study the quantitative Glanser property in the context of maps between tori of differing dimension instead of as a (semi-)group action. We also only consider matrices with entries being non-constant polynomials evaluated at primes, extending on the work of Velani and Nair, and Bulinski and Fish to a more general setting.
Paper Structure (4 sections, 10 theorems, 37 equations)

This paper contains 4 sections, 10 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. Proof of theorem \ref{['theorem1']}
  3. Proof of theorem \ref{['theorem2']}
  4. Acknowledgements

Key Result

Theorem 1.3

Let $P$ be the set of primes. Fix $\delta > 0$ and some $f \in \mathbb{Q}[x]$ where $f$ is a non-constant polynomial of degree $L \geq 1$ with $f(\mathbb{Z}) \subseteq \mathbb{Z}$. Then there exists some $\epsilon_0 = \epsilon_0(f, \delta)$ such that with $\mathcal{M} = f(P)$, $(\mathbb{T}, \mathbb{

Theorems & Definitions (14)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Definition 2.4
  • ...and 4 more