Sum of elements in finite Sidon sets II

2022/05/02 by Ding, Yuchen
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2205.01084

Abstract

A set S⊂\1,2,...,n\ is called a Sidon set if all the sums a+b~~(a,b∈ 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 ∑_{a∈ S,~|S|=Sn}a=(1)/(2)n^3/2+O(n^111/80+ε), where ε>0 is an arbitrary small constant. In this note, we give an extension of the above formula. We show that ∑_{a∈ S,~|S|=Sn}a^ℓ=(1)/(ℓ+1)n^ℓ+1/2+O(n^ℓ+61/160) for any positive integers ℓ. 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.

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