# Generalization of a theorem of Erdos and Renyi on Sidon Sequences - Mathematics > Number Theory

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Abstract: Erd\H os and R\-{e}nyi claimed and Vu proved that for all $h \ge 2$ and forall $\epsilon > 0$, there exists $g = g h\epsilon$ 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 $|A \cap 1,x| \gg x^{1-h- \epsilon}$.We give two new proofs of this result. The first one consists of an explicitconstruction of such a sequence. The second one is probabilistic and shows theexistence of such a $g$ that satisfies $g h\epsilon \ll \epsilon^{-1}$,improving the bound $g h\epsilon \ll \epsilon^{-h+1}$ obtained by Vu.Finally we use the -alteration method- to get a better bound for$g 3\epsilon$, obtaining a more precise estimate for the growth of $B 3g$sequences.

Author: Javier Cilleruelo, Sandor Z. Kiss, Imre Z. Ruzsa, Carlos Vinuesa

Source: https://arxiv.org/