Generalization of a theorem of Erdos and Renyi on Sidon Sequences

TL;DR

This paper provides new proofs for a theorem on Sidon sequences by Erdős and Rényi, including explicit and probabilistic constructions, and improves bounds on the number of representations in such sequences.

Contribution

It introduces two new proofs of the theorem, one explicit and one probabilistic, and refines bounds on the growth of sequences with bounded representations.

Findings

Explicit construction of sequences with bounded representations.

Probabilistic proof showing improved bounds on g_h(ε).

Enhanced bounds for g_3(ε) using the alteration method.

Abstract

Erd\H os and R\'{e}nyi claimed and Vu proved that for all $h \ge 2$ and for all $\epsilon > 0$, 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 explicit construction of such a sequence. The second one is probabilistic and shows the existence 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 $B_3[g]$ sequences.