# Probability distribution from equidistribution - II

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 $$|\sqrt{x^2+y^2}| where $$(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?