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Szemerédi–Trotter type theorem in finite field
This question is about the content of this paper by J. Bourgain, N. Katz, T. Tao.
In the final step (page 18) of the proof of Szemerédi-Trotter type theorem, we have already known
$$|A''+A''|\lesssim ...
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Why should it be hard to generalize Dvir's proof of the finite field Kakeya conjecture to the Euclidean case?
Let $q$ be prime and let
$q\delta \sim 1.$ Let $K$ be any set of $C_n\delta$-separated tubes in $B(0,2)$, where $C_n$ is some constant depending on $n$. Let us consider a grid of $q^n$ points scaled ...