Generalized Sidon sets of perfect powers

TL;DR

This paper demonstrates the existence of a dense set of perfect powers with bounded representation counts for sums of h elements, using probabilistic techniques to extend classical additive number theory results.

Contribution

It introduces a probabilistic construction of dense perfect power sets with bounded sum representations, advancing understanding of generalized Sidon sets.

Findings

Existence of dense perfect power sets with bounded sum solutions

Application of probabilistic methods in additive number theory

Extension of Sidon set concepts to perfect powers

Abstract

For $h \ge 2$ and an infinite set of positive integers $A$, let $R_{A,h}(n)$ denote the number of solutions of the equation $a_{1} + a_{2} + \dots{} + a_{h} = n, a_{1} \in A, \dots{} ,a_{h} \in A, a_{1} < a_{2} < \dots{} < a_{h}.$ In this paper we prove the existence of a set $A$ formed by perfect powers with almost possible maximal density such that $R_{A,h}(n)$ is bounded by using probabilistic methods.