The $m$-th Element of a Sidon Set

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 $\varepsilon >0$, we have $$\sum_{a\in S} a = \frac 12 n^{3/2} + \mathcal O \left (n^{11/8} \right )$$ for all $n\le N$ but at most $\mathcal O_{\varepsilon} \left (N^{\frac 45 + \varepsilon} \right )$ exceptions.