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 $t(a,b)\equiv(x,y)\bmod p$. 


1. 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 $t(a,b,c)\equiv(x,y,z)\bmod p$.


2. 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 $\lceil p^{1/3}\rceil$?



Negative Dirichlet Pigeonhole Principle
No. Observe that $a/b=x/y$, from which it follows that $ay=xb$. Since $a,b,x,y$ are all in $(0,\sqrt{p})$, it follows that the equality $ay=xb$ holds not merely mod $p$, but in $\mathbb{Z}$. Since $a,b$ are coprime, it follows that $a\mid x$ and $b\mid y$. Hence, $\lvert x\rvert\geq \lvert a\rvert$ and $\lvert y\rvert\geq \lvert b\rvert$.


