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?



Arithmetic progressions modulo $p$ under the squaring map
This is a variant of a common theme. It should follow from more or less standard exponential sums estimates. The general buzzword is Erdos-Turan inequality. The answer should be yes and it might follow from the results of:


A. Granville, I. E. Shparlinski and A. Zaharescu, On the
distribution of rational functions along a curve over $\mathbb{F}\_p$ and residue races, J. Number Theory, 112 (2005), 216--237.


or C. Cobeli and A. Zaharescu, On the distribution of the $\mathbb{F}\_p$-points on an affine curve in $r$ dimensions, Acta Arithmetica 99 (2001), 321--329.


Or perhaps even earlier papers.


