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Pick a random pair $(a,b)\in\mathbb Z_n^2\backslash\{0,0\}$. 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 $k=O(1)$ random pairs $(a_1,b_1),(a_2,b_2),\dots,(a_k,b_k)\in\mathbb Z_n^2\backslash\{0,0\}$ where $k\geq3$. What is the probability that $$\frac{\max(N_2(a_1,b_1,n),N_2(a_2,b_2,n),\dots,N_2(a_k,b_k,n))}{\sqrt n}<\frac{n^\epsilon}{\sqrt n}?$$

What I want validation is on following possibility:

  1. Can we get $k=O(1)$ with $k\geq3$ such coprime pairs of $(a_1,b_1),(a_2,b_2),\dots,(a_k,b_k)$ for all primes $n$ large enough with $$\sqrt n<a_1,b_1,a_2,b_2,\dots,a_k,b_k<n-\sqrt n$$ and with $$\frac{\max(N_2(a_1,b_1,n),N_2(a_2,b_2,n),\dots,N_2(a_k,b_k,n))}{\sqrt n}<\frac{n^\epsilon}{\sqrt n}?$$

  2. Slightly different version is can we get $2k=O(1)$ with $k\geq3$ such mutually coprime integers $a_1,b_1,a_2,b_2,\dots,a_k,b_k$ for all primes $n$ large enough with $$\sqrt n<a_1,b_1,a_2,b_2,\dots,a_k,b_k<n-\sqrt n$$ and with $$\frac{N_2(a_1,b_1,a_2,b_2,\dots,a_k,b_k,n)}{\sqrt n}<\frac{n^\epsilon}{\sqrt n}$$ where $N_2(a_1,b_1,a_2,b_2,\dots,a_k,b_k,n)$ is direct generalization of $N_2(a,b,n)$.

There is a single $t$ in 2. while three different $t$'s are allowed in 1.

I only need the result at $k=3$.

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