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Generalization of a theorem of Erdős and Rényi on Sidon sequences

AuthorsCilleruelo, Javier; Kiss, Sándor Z.; Ruzsa, Imre Z.; Vinuesa, Carlos
KeywordsSidon sequences
The probabilistic method
Issue Date2010
CitationRandom Structures and Algorithms 37(4): 455-464 (2010)
AbstractErdős 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] sequences
Description10 páginas.
Publisher version (URL)http://dx.doi.org/10.1002/rsa.20350
Appears in Collections:(ICMAT) Artículos
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