can be very useful, as we will see in the proof of Theorem \[thm-newsumproduct\] which uses this fact to prove certain sum-product estimates. The existence of such a low-degree hypersurface containing most of the curves can also be obtained when there are many $r$-rich curves of bounded degree with ‘two degrees of freedom’ i.e. through every pair of points there are at most $O(1)$ curves (see Remark \[rem-bounded-degree-curves\]).

#### Counting arithmetic progressions {#counting-arithmetic-progressions .unnumbered}

An $r$-term arithmetic progression in $\C^d$ is simply a set of $r$ points of the form $\{y,y+x,y+2x, \ldots,y+(r-1)x\}$ with $x,y \in \C^d$. This is a special case of $r$ collinear points and, for this case, we can derive a tighter bound than for the general case. In a nutshell, we can show that a $d$-dimensional grid contains the largest number of $r$-term progressions, among all sets that do not contain a large $d-1$ dimensional subset. The main extra property of arithmetic progressions we use in the proof is that they behave well under products. That is, if we take a Cartesian product of $V$ with itself, the number of progressions of length $r$ squares.

For a finite set $V \subset \C^d$, let us denote the number 