Let $p$ be large prime in $[T,2T]$ where $T$ is a parameter.

1. Can we have an integer pair $a,b$ satisfying $ab\equiv c\bmod p$ such that $|c|$ is of size $O(\operatorname{polylog}(T))$ and $a,b$ are of size $O(T^{1/2 +\varepsilon})$?

2. If $1/2$ is impossible what is the best rational in exponent we can get?

3. Given $p$ how many such minimal generically positioned pairs are possible?