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 $0<t<n$.

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 $\sqrt n<a,b<n-\sqrt n$ with $N_2(a,b,n)/\sqrt n<n^\epsilon/\sqrt n$.

Pick a random pair $(a,b)\in\mathbb Z_n^2\setminus\{0,0\}$ and pick another prime $n'$ with $n<n'<2n$. 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 $0<t<n$ 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 $0<t'<n'$.

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 $n<m<2n$ holds)?

It looks like given $\epsilon>0$ and primes $n,m$ large enough and randomly taken with bounds $n<m<2m$ there should be coprime $a,b$ with $\sqrt m<a,b<n-\sqrt m$ 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.