Sets of recurrence as bases for the positive integers

TL;DR

This paper investigates sets defined by polynomial fractional parts and their ability to generate all positive integers through finite sums, revealing degree-dependent basis properties.

Contribution

It characterizes when sets of the form involving polynomial fractional parts form bases of finite order, especially distinguishing between quadratic and higher degrees.

Findings

Sets with degree ≥ 3 are generically bases of order 2.

Quadratic polynomial sets are not bases, but their sumsets have density 1.

Provides conditions under which these sets cover all positive integers through finite sums.

Abstract

We study sets of the form $A = \big\{ n \in \mathbb N \big| \lVert p(n) \rVert_{\mathbb R / \mathbb Z} \leq \varepsilon(n) \big\}$ for various real valued polynomials $p$ and decay rates $\varepsilon$. In particular, we ask when such sets are bases of finite order for the positive integers. We show that generically, $A$ is a basis of order $2$ when $\operatorname{deg} p \geq 3$, but not when $\operatorname{deg} p = 2$, although then $A + A$ still has asymptotic density $1$.