Diophantine Tuples over ${\mathbb{Z}}_p$

For an element $r$ of a ring $R$, a Diophantine $D(r)$ $m$-tuple is an $m$-tuple $(a_1,a_2,\dots ,a_m)$ of elements of $R$ such that for all $i,j$ with $i\mathrm{\ne }j$, $a_i{a}_j+r$ is a perfect square in $R$. In this article, we compute and estimate the measures of the sets of $D(r)$ $m$-tuples in the ring ${\mathbb{Z}}_p$ of $p$-adic integers, as well as its residue field ${\mathbb{F}}_p$.