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Sets of recurrence as bases for the positive integers

Volume 174 / 2016

Jakub Konieczny Acta Arithmetica 174 (2016), 309-338 MSC: Primary 11J54; Secondary 11P99. DOI: 10.4064/aa8125-4-2016 Published online: 12 July 2016

Abstract

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 $1$.

Authors

  • Jakub KoniecznyMathematical Institute
    University of Oxford
    Andrew Wiles Building
    Radcliffe Observatory Quarter
    Woodstock Road
    Oxford, OX2 6GG, U.K.
    e-mail

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