Abstract
In this paper, we study (random) sequences of pseudo s-th powers, as introduced by Erdős and Rényi (Acta Arith 6:83–110, 1960). Goguel (J Reine Angew Math 278/279:63–77, 1975) proved that such a sequence is almost surely not an asymptotic basis of order s. Our first result asserts that it is however almost surely a basis of order \(s+\epsilon \) for any \(\epsilon >0\). We then study the s-fold sumset \(sA=A+\cdots +A\) (s times) and in particular the minimal size of an additive complement, that is a set B such that \(sA+B\) contains all large enough integers. With respect to this problem, we prove quite precise theorems which are tantamount to asserting that a threshold phenomenon occurs.
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The first author was supported by Grants MTM 2011-22851 of MICINN and ICMAT Severo Ochoa Project SEV-2011-0087. The second, third and fourth author were supported by an ANR Grant Cæsar, Number ANR 12 - BS01 - 0011. All the authors are thankful to École Polytechnique for making possible their collaboration.
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Cilleruelo, J., Deshouillers, JM., Lambert, V. et al. Additive properties of sequences of pseudo s-th powers. Math. Z. 284, 175–193 (2016). https://doi.org/10.1007/s00209-016-1651-8
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DOI: https://doi.org/10.1007/s00209-016-1651-8