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arXiv · 1504.02410

Sets of recurrence as bases for the positive integers

Jakub Konieczny
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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 $1$.

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Jakub Konieczny. 2018-07-19. Sets of recurrence as bases for the positive integers. https://doi.org/10.4064/aa8125-4-2016

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