Sets of recurrence as bases for the positive integers

Abstract

We study sets of the form $A=\{n\in \mathbb{N}|\Vert p(n){\Vert }_{\mathbb{R}/\mathbb{Z}}\le \epsilon (n)\}$ for various real valued polynomials $p$ and decay rates $\epsilon$. 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 $\mathrm{deg}p\ge 3$, but not when $\mathrm{deg}p=2$, although then $A+A$ still has asymptotic density $1$.

26 pages

References in corpus (1)

• The quantitative behaviour of polynomial orbits on nilmanifolds

Cited by in corpus (1)

• Weakly mixing sets and polynomial equations