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?



Small near-reciprocals
There is a nice article of Heath-Brown in the Mathematical Intelligencer, called Arithmetic applications of Kloosterman sums, where this problem is discussed under the heading, "An elementary problem." He derives exponent $3/4$ and remarks that it is open to improve on it. I am not aware of any subsequent improvements.


