Sets of recurrence as bases for the positive integers

Jakub Konieczny¹

¹ Mathematical Institute University of Oxford Andrew Wiles Building Radcliffe Observatory Quarter Woodstock Road Oxford, OX2 6GG, U.K.

Acta Arithmetica, Tome 174 (2016) no. 4, pp. 309-338

Résumé

We study sets of the form $\mathcal{A} = \{ n \in \mathbb {N} \mid \|{p(n)}\| \leq \varepsilon(n) \}$ 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, $\mathcal A$ is a basis of order 2 when $\deg p \geq 3$, but not when $\deg p = 2$, although then $\mathcal A + \mathcal A$ still has asymptotic density $1$.

Détail

DOI : 10.4064/aa8125-4-2016

Keywords: study sets form mathcal mathbb mid leq varepsilon various real valued polynomials decay rates varepsilon particular ask sets bases finite order positive integers generically mathcal basis order nbsp deg geq deg although mathcal mathcal still has asymptotic density nbsp