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Generalization of a theorem of Erdős and Rényi on Sidon sequences. (English) Zbl 1226.11018

Given an integer \(h\geq 2\) we say that a sequence of integers \(A\) is a \(B_h[g]\) sequence if every integer \(n\) has at most \(g\) representations as a sum of \(h\) elements of \(A\). Improving results of Van H. Vu [Duke Math. J. 105, No.1, 107–134 (2000; Zbl 1013.11063)] the authors prove that for any \(\varepsilon>0\) and \(h\geq 2\), there exists \(g=g_h(\varepsilon)\ll\varepsilon^{-1}\) and a \(B_h[g]\) sequence \(A\), such that \(A(x)\gg x^{\frac{1}{h}-\varepsilon}\), where \(A(x)=|A\cap [1,x]|\). For every \(\delta>0\) and for every \(g\geq 1\) there is a \(B_3[g]\) sequence \(A\), such that \(A(x)\gg x^{\frac{g}{3g+2}-\delta}\). They use the probabilistic method.

MSC:

11B13 Additive bases, including sumsets
11B05 Density, gaps, topology

Citations:

Zbl 1013.11063
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References:

[1] Alon, The probabilistic method (2000)
[2] Cilleruelo, Probabilistic constructions of B2[g] sequences, Acta Math Sin 26 pp 1309– (2010) · Zbl 1241.11025
[3] Cilleruelo, Sidon sets in \(\mathbb{N}\)d, J Combin Theory Ser A 117 pp 857– (2010) · Zbl 1246.11022
[4] Erdos, Additive properties of random sequences of positive integers, Acta Arith 6 pp 33– (1960)
[5] Erdos, Representations of integers as the sum of k terms, Random Struct Algorithms 1 pp 245– (1990)
[6] Halberstam, Sequences (1983) · doi:10.1007/978-1-4613-8227-0
[7] Vu, On a refinement of Waring’s problem, Duke Math J 105 pp 107– (2000) · Zbl 1013.11063
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