of $r$-term arithmetic progressions contained in $V$ by $\AP_r(V)$. We first observe that, for all sufficiently small $r$, the grid $G_d$ (defined above) contains at least $\gtrsim_d n^2/r^d$ $r$-term progressions. To see where the extra factor of $r$ comes from, notice that the $2r$-rich lines in $G_d$ will contain $r$ arithmetic progressions of length $r$ each. Our main theorem shows that this is optimal, as long as there is no large low-dimensional set.

\[thm-apbound\] Let $0 < \epsilon < 1$ and $V \subset \C^d$ be a set of size $n$ and suppose that for some $r \geq 4$ we have $$\AP_r(V) \gg_{d,\epsilon} \frac{n^2}{r^{d-{\epsilon}}}.$$ Then, there exists a subset $\tilde V \subset V$ of size $\gtrsim_{d,\epsilon} \frac{n}{r^{{2d/{\epsilon}}-1}}$ contained in a hyperplane.

Related Work
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Using the incidence bound between points and lines in $\R^3$ proved in [@GK10], one can prove the following theorem from which Conjecture \[conj-rich\] in $\R^3$ trivially follows (see Appendix \[sec-stronger-thm1\]).

\[thm-conj-d=3\] Given a set $V$ of $n$ points in $\R^3$, let $s_2$ denote the maximum number of points of $V$ contained in a $2$-flat. Then for $r\ge 2$, $$|\L_r(V)|\lesssim \frac{n^2}{r^4}+\frac{ns_2}{r^3}+\frac{n}{r}.$$

Similarly, using the results of in [@SS14], we can prove the following theorem from which a slightly 