Generalization of a theorem of Erdős and Rényi on Sidon sequences

10 páginas.Erdős and Rényi claimed and Vu proved that for all h ≥ 2 and for all ϵ > 0, there exists g = gh(ϵ) and a sequence of integers A such that the number of ordered representations of any number as a sum of h elements of A is bounded by g, and such that |A ∩ [1,x]| ≫ x1/h-ϵ. We give two new proofs of this result. The first one consists of an explicit construction of such a sequence. The second one is probabilistic and shows the existence of such a g that satisfies gh(ϵ) ≪ ϵ−1, improving the bound gh(ϵ) ≪ ϵ−h+1 obtained by Vu. Finally we use the “alteration method” to get a better bound for g3(ϵ), obtaining a more precise estimate for the growth of B3[g] sequencesSupported by MICINN (MTM 2008-03880); ERC–AdG (228005); Hungarian National Foundation for Scientific Research (OTKA) (K 61908, K 72731).Peer reviewe