Let $p$ be a large prime. I would like to say that the multi-set $[1,p^{1-\varepsilon}]^2 = \{ab \mod p: a, b \in [1,p^{1-\varepsilon}]\}$ is close to uniformly distributed, i.e. that every nonzero residue class mod $p$ occurs with almost equal frequency. 


Quantitatively, is it true that (almost?) every nonzero residue class appears at most $O(p^{1-2\varepsilon})$ times?



Well-distribution of square of an interval $[1,p^{1-\varepsilon}]$ modulo a prime $p$
Here is a more explicit version of Noam Elkies's comment.


**Theorem.** Let $p>2$ be a prime number. Let $\mathcal{U},\mathcal{V}\subseteq\{1,2,\dots,p-1\}$ be two intervals, and let $r\in\{1,2,\dots,p-1\}$
be a nonzero residue modulo $p$. Then
$$\Biggl|\sum\_{\substack{u\in\mathcal{U},\ v\in\mathcal{V} \\ uv\equiv r \pmod{p}}} 1-\frac{|\mathcal{U}||\mathcal{V}|}{p-1}\Biggr|<2p^{1/2}(\log p)^2.$$


You can find a proof from scratch [here](http://www.renyi.hu/~gharcos/weil.pdf) (see Theorem 9).


