  Green used an arithmetic analogue of Szemer\'edi's celebrated regularity
lemma to prove the following strengthening of Roth's theorem in vector spaces.
For every $\alpha>0$, $\beta<\alpha^3$, and prime number $p$, there is a least
positive integer $n_p(\alpha,\beta)$ such that if $n \geq n_p(\alpha,\beta)$,
then for every subset of $\mathbb{F}_p^n$ of density at least $\alpha$ there is
a nonzero $d$ for which the density of three-term arithmetic progressions with
common difference $d$ is at least $\beta$. We determine for $p \geq 19$ the
tower height of $n_p(\alpha,\beta)$ up to an absolute constant factor and an
additive term depending only on $p$. In particular, if we want half the random
bound (so $\beta=\alpha^3/2$), then the dimension $n$ required is a tower of
twos of height $\Theta \left((\log p) \log \log (1/\alpha)\right)$. It turns
out that the tower height in general takes on a different form in several
different regions of $\alpha$ and $\beta$, and different arguments are used
both in the upper and lower bounds to handle these cases.
