Generalization of a theorem of Erdos and Rényi on Sidon sequences

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]| À 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.