From Dirichlet Pigeonhole Principle if $p$ is a prime and if $a,b\in\mathbb Z$ are in $(0,p/2)$ then there is a $t\in(0,p)\cap\mathbb Z$ such that $\|(x,y)\|_\infty<\lceil\sqrt p\rceil$ holds where $t(a,b)\equiv(x,y)\bmod p$.

- Is there no distinct coprime $a,b$ in $(0,\lceil\sqrt p\rceil)$ such that there is $t\in(0,p)\cap\mathbb Z$ with $\|(x,y)\|_\infty<\|(a,b)\|_\infty$?

From Dirichlet Pigeonhole Principle if $p$ is a prime and if $a,b,c\in\mathbb Z$ are in $(0,p/2)$ then there is a $t\in(0,p)\cap\mathbb Z$ such that $\|(x,y,z)\|_\infty<\lceil p^{2/3}\rceil$ holds where $t(a,b,c)\equiv(x,y,z)\bmod p$.

- Is there no distinct pairwise coprime $a,b,c$ in $(0,\lceil p^{1/3}\rceil)$ such that there is $t\in(0,p)\cap\mathbb Z$ with $\|(x,y,z)\|_\infty<\|(a,b,c)\|_\infty$? Is it possible to increase size of $a,b,c$ to a larger value than $\lceil p^{1/3}\rceil$?