Pick a random pair $(a,b)\in\mathbb Z_n^2\backslash\{0,0\}$. Denote $N_r(a,b)$ to be minimum $\ell_r$ 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 is it true as $n\rightarrow\infty$ the distribution of $N_2(a,b)/\sqrt{n}$ coincides with distribution of $1/\sqrt y$ where $x+iy$ is picked at random with respect to hyperbolic measure from $\{z:|z|\geq1,|\mathcal R(z)|\leq\frac12\}$?

1. This it is true when $n$ is prime 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' which he says follows from an equidistribution result. However how to see this at least and at least for what fraction of composites is this true?

2. What about for $r=\infty$?

Pick a random pair $(a,b)\in\mathbb Z_n^2\backslash\{0,0\}$. Denote $N_r(a,b)$ to be minimum $\ell_r$ 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 is it true as $n\rightarrow\infty$ the distribution of $N_2(a,b)/\sqrt{n}$ coincides with distribution of $1/\sqrt y$ where $x+iy$ is picked at random with respect to hyperbolic measure from $\{z:|z|\geq1,|\mathcal R(z)|\leq\frac12\}$?

This is true when $n$ is prime 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' which he says follows from an equidistribution result.

1. However how to see this?

2. Is there an analogous result for semi primes?

3. What fraction of composites have analog?

4. What about for $r=\infty$?