The $m$-th Element of a Sidon Set

Abstract

We prove that if $A=\{a_1,\dots ,a_{|A|}\}\subset \{1,2,\dots ,n\}$ is a Sidon set so that $|A|=n^{1/2}-L^{\prime }$, then $$a_m=m\cdot n^{1/2}+\mathcal{O}\left(n^{7/8}\right)+\mathcal{O}\left(L^{1/2}\cdot n^{3/4}\right)$$ where $L=\max \{0,L^{\prime }\}$. As an application of this, we give easy proofs of some previously derived results. We proceed on to proving that for a dense Sidon set $S$ and for any $\epsilon >0$, we have $$\sum _{a\in S}a=\frac{1}{2}{n}^{3/2}+\mathcal{O}\left(n^{11/8}\right)$$ for all $n\le N$ but at most ${\mathcal{O}}_{\epsilon }\left(N^{\frac{4}{5}+\epsilon }\right)$ exceptions.