# Probability distribution from standard domain (two primes) - IV

Pick a random pair $$(a,b)\in\mathbb Z_n^2\setminus\{0,0\}$$. Denote $$N_2(a,b,n)$$ 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 $$t\in\mathbb Z$$ with $$0.

This is the result from Akshay Venkatesh when $$n$$ is prime. Then it is true as $$n\rightarrow\infty$$ the distribution of $$N_2(a,b,n)/\sqrt{n}$$ coincides with distribution of $$1/\sqrt y$$ where $$x+iy$$ is picked at random with respect to hyperbolic measure from $$\{z:|z|\geq1,|\mathcal R(z)|\leq\frac12\}$$. Hence It looks like given $$\epsilon>0$$ and prime $$n$$ large enough and randomly taken there should be coprime $$a,b$$ with $$\sqrt n with $$N_2(a,b,n)/\sqrt n.

Pick a random pair $$(a,b)\in\mathbb Z_n^2\setminus\{0,0\}$$ and pick another prime $$n'$$ with $$n. Denote $$N_2(a,b,n,n')$$ to be sum of minimum $$\ell_2$$ norms of vector $$(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$$ with $$0 and of minimum $$\ell_2$$ norms of vector $$(x',y')$$ as $$(x',y')$$ ranges over all non-zero integral solutions to $$(x',y')\equiv t'(a,b)\bmod n'$$ where $$t'\in\mathbb Z$$ with $$0.

What is the distribution of $$N_2(a,b,n,m)/(\sqrt n+\sqrt m)$$ when $$n,m$$ are independent primes as $$n+m\rightarrow\infty$$ (preferably when $$n holds)?

It looks like given $$\epsilon>0$$ and primes $$n,m$$ large enough and randomly taken with bounds $$n there should be coprime $$a,b$$ with $$\sqrt m with $$N_2(a,b,n,m)/(\sqrt n+\sqrt m)<(n+m)^\epsilon/(\sqrt n+\sqrt m)$$. This is what I want to ascertain.