Pick a random pair $(a,b)\in\mathbb Z_n^2\backslash\{0,0\}$. Denote $N(a,b)$ to be minimum $\ell_2$ norm of vector $(x,y)$ as $(x,y)$ ranges over all non-zero integral solutions to $(x,y)\equiv t(a,b)\bmod n$ where $t\in\mathbb Z$. Then if $n$ is prime it is true that as $n\rightarrow\infty$ the distribution of $N(a,b)/\sqrt{n}$ coincides with distribution of $1/\sqrt y$ where $x+iy$ is picked at random with respect to hyperbolic measure from $\{z:|z|\geq1,|\mathcal R(z)|\leq\frac12\}$ (from Akshay Venkatesh's paper 'Spectral theory of automorphic forms, a very brief introduction' published in 'Equidistribution in Number Theory, An Introduction edited by Andrew Granville, Zeév Rudnick').

Given a prime $p$ and an $\epsilon\in(0,1/\log p)$ suppose there is a $t$ such that for a given $(a,b)\in\mathbb Z^2\backslash\{0,0\}$ we have $|\sqrt{x^2+y^2}|<p^\epsilon$ where $(x,y)\equiv t(a,b)\bmod p$ holds how many other primes can we find on an average such that there are $t$ with corresponding $(x,y)$ having $\ell_2$ norm bound by $p^\epsilon$ at the same $(a,b)$? Can we find $p^{O(\epsilon)}$ such primes between $p$ and $2p$?

Pick a random pair $(a,b)\in\mathbb Z_n^2\backslash\{0,0\}$ and denote $N(a,b)$ to be minimum $\ell_2$ norm of vector $(x,y)$ as $(x,y)$ ranges over all non-zero integral solutions to $(x,y)\equiv t(a,b)\bmod n$ where $t\in\mathbb Z$. Then if $n$ is prime it is true that as $n\rightarrow\infty$ the distribution of $N(a,b)/\sqrt{n}$ coincides with distribution of $1/\sqrt y$ where $x+iy$ is picked at random with respect to hyperbolic measure from $\{z:|z|\geq1,|\mathcal R(z)|\leq\frac12\}$ (from Akshay Venkatesh's paper 'Spectral theory of automorphic forms, a very brief introduction' published in 'Equidistribution in Number Theory, An Introduction edited by Andrew Granville, Zeév Rudnick').

Given a prime $p$ and an $\epsilon\in(0,1/\log p)$ suppose there is a $t$ such that for a given $(a,b)\in\mathbb Z^2\backslash\{0,0\}$ we have $|\sqrt{x^2+y^2}|<p^\epsilon$ where $(x,y)\equiv t(a,b)\bmod p$ holds how many other primes can we find on an average such that there are $t$ with corresponding $(x,y)$ having $\ell_2$ norm bound by $p^\epsilon$ at the same $(a,b)$? Can we find $p^{O(\epsilon)}$ such primes between $p/2$ and $p$ and in particular can we find at least one additional prime?