The $m$-th Element of a Sidon Set

We prove that if $A=\{a_1,\dots ,a_{\mathrm{\mid }A\mathrm{\mid }}\}\subset \{1,2,\dots ,n\}$ is a Sidon set so that $\mathrm{\mid }A\mathrm{\mid }=n^{1\mathrm{/}2}-L^{\mathrm{\prime }}$, then $$a_m=m\cdot n^{1\mathrm{/}2}+\mathcal{O}\left(n^{7\mathrm{/}8}\right)+\mathcal{O}(L^{1\mathrm{/}2}\cdot n^{3\mathrm{/}4})$$ where $L=\max \{0,L^{\mathrm{\prime }}\}$. 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 $$\sum _{a\in S}a=\frac{1}{2}{n}^{3\mathrm{/}2}+\mathcal{O}\left(n^{11\mathrm{/}8}\right)$$ for all $n\le N$ but at most ${\mathcal{O}}_{\epsilon }\left(N^{\frac{4}{5}+\epsilon }\right)$ exceptions.