Sum of elements in finite Sidon sets II

TL;DR

This paper extends the asymptotic analysis of sum of elements in maximum Sidon sets within {1,...,n}, providing formulas for higher powers and other summations, especially when the set size is near n^{1/2}.

Contribution

It generalizes previous results by deriving asymptotic formulas for sums of powers and other functions over Sidon sets with size close to n^{1/2}.

Findings

Derived asymptotic formulas for sums of powers of Sidon set elements.

Extended analysis to Sidon sets with size near n^{1/2}.

Provided bounds for various summation types involving Sidon sets.

Abstract

A set $S\subset\{1,2,...,n\}$ is called a Sidon set if all the sums $a+b~~(a,b\in S)$ are different. Let $S_n$ be the largest cardinality of the Sidon sets in $\{1,2,...,n\}$. In a former article, the author proved the following asymptotic formula $$\sum_{a\in S,~|S|=S_n}a=\frac{1}{2}n^{3/2}+O(n^{111/80+\varepsilon}),$$ where $\varepsilon>0$ is an arbitrary small constant. In this note, we give an extension of the above formula. We show that $$\sum_{a\in S,~|S|=S_n}a^{\ell}=\frac{1}{\ell+1}n^{\ell+1/2}+O\left(n^{\ell+61/160}\right)$$ for any positive integers $\ell$. Besides, we also consider the asymptotic formulae of other type summations involving Sidon sets. The proofs are established in a more general setting, namely we obtain the asymptotic formulae of the Sidon sets with $t$ elements when $t$ is near the magnitude $n^{1/2}$.