I feel that the following problem should be known, but I'm not sure where to look for it.

Fix a real constant $\frac{1}{2} \ge \epsilon > 0$. For varying primes $p$, Let $A_p$ denote the set of residue classes coming from the first $\lfloor p \epsilon \rfloor$ integers. Let $B_p$ denote the squares (modulo $p$) of the elements of $A_p$. Then one might ask whether $$\lim_{p \rightarrow \infty} \frac{|A_p \cap B_p|}{|A_p|} =^{?} \epsilon.$$ It's true for $\epsilon = \frac{1}{2}$, but that's a degenerate case where $B_p$ can essentially be replaced by $\mathbf{F}^{\times 2}_p$, in which case the answer follows from any non-trivial upper bound on character sums (say the Polya-Vinagradov inequality). Is it true more generally?