many points of the configuration (by finding a ‘flat’ point of the surface).

Application: Sum-product estimates
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Here, we show a simple application of our techniques to prove sum product estimates over $\C$. The estimates we will get can also be derived from the Szemerédi-Trotter theorem in the complex plane (see Section \[sec-newsumproduct-usingST\]) and we include them only as an example of how to use a higher dimensional theorem in this setting. We hope that future progress on proving Conjecture \[conj-rich\] will result in progress on sum product problems.

We begin with some notations. For two sets $A,B \subset \C$ we denote by $A+B = \{a+b \,|\, a,b \in A\}$ the sum set of $A$ and $B$. For a set $A \subset \C$ and a complex number $t \in \C$ we denote by $tA = \{ ta \,|\, a \in A \}$ the dilate of $A$ by $t$. Hence we have that $A+tA = \{ a + ta'\,|\, a,a' \in A\}$.

\[thm-newsumproduct\] Let $A \subset \C$ be a set of $N$ complex numbers and let $1\ll C \ll \sqrt{N}$. Define the set $$T_C = \left\{ t \in \C \,\left|\, |A + tA| \leq \frac{N^{1.5}}{C\sqrt{\log N}} \right. \right\}.$$ Then, $ |T_C| 