Improved upper bounds on Diophantine tuples with the property $D(n)$

Let $n$ be a non-zero integer. A set $S$ of positive integers is a Diophantine tuple with the property $D(n)$ if $ab+n$ is a perfect square for each $a,b\in S$ with $a\mathrm{\ne }b$. It is of special interest to estimate the quantity $M_n$, the maximum size of a Diophantine tuple with the property $D(n)$. In this notes, we show the contribution of intermediate elements is $O(\log \log \mathrm{\mid }n\mathrm{\mid })$, improving a result by Dujella. As a consequence, we deduce that $M_n\le (2+o(1))\log \mathrm{\mid }n\mathrm{\mid }$, improving the best-known upper bound on $M_n$ by Becker and Murty.