Shifts of multiplicative subgroup of a field 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 $|H\cap (H+c_1)\cap (H+c_2)\cap\cdots\cap (H+c_m)|$?
 A: In fields of prime order, factoring the difference of squares shows that the number of solutions to $x^2=y^2+c$ is what you expect it to be, from which it follows the intersection $H\cap (H+c)$ has size about what you expect. For triple intersections such as $H\cap (H+1)\cap (H+2)$, one there is an elementary argument for certain values of $p$, but not others (in this example, for $p=3,7\pmod 8$). In general you will need estimate the character sum $$\sum_x \chi(x(x+c_1)(x+c_2)\dotsb(x+c_m))$$ where $\chi$ is the Legendre symbol modulo $p$. This can be estimated by invoking Riemann hypothesis for curves.
By the way, if $H$ were a small subgroup of $ F* $ (of size $n^{1-\epsilon}$), then the sum-product estimates show that the intersection $|H\cap (H+c)|$ can never be large. See for example, this paper for a related problem of estimating the product of $H$ and $H+c$. 
The following is wrong since (H+c) is not closed under multiplication:
One can get the result for the intersection by using Ruzsa's triangle inequality. Namely, $$|H(H+c)|\leq \frac{|H(H\cap (H+c))||(H+c)(H\cap (H+c))|}{|(H\cap (H+c))|}\leq \frac{|H|^2}{|(H\cap (H+c))|}$$.
A: Fix a positive integer $e$, let $n$ be a prime power congruent to 1 modulo $e$, let $F$ be the field of $n$ elements, let $H$ be the subgroup of index $e$ in the multiplicative group of $F$. Then the study of the size of $H\cap(H+c)$ is called $\it cyclotomy$, it goes back to Gauss (for the cases $e=2,3,4$ and $n$ a prime), there's a book by Tom Storer, and many papers. 
