Let $F$ be a finite field of odd order n. Let $H$ be the subgroup of $F^\times$ composed of the nonzero squares. What can we say about $|H\cap (H+c)|$ given $c\in F$? The expected value of $|H\cap (H+c)|$ is roughly $n/4$ when $c$ is random. But I wonder what the distribution looks like. For example, Is there an infinite family of finite fields for which $|H\cap (H+c)|$ deviates from its expectation dramatically, say by a constant multiplicative factor?
More generally, how about the intersection of a multiple number of additive shifts, like $|H\cap (H+c_1)\cap (H+c_2)\cap\cdots\cap (H+c_m)|$?