Szemerédi–Trotter type theorem in finite field

Question

This question is about the content of this paper by J. Bourgain, N. Katz, T. Tao.

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In the final step (page 18) of the proof of Szemerédi-Trotter type theorem, we have already known

$$|A''+A''|\lesssim N^{\frac{1}{2}+C\epsilon}$$

Similarly, by the Balog-Szemerédi-Gowers Theorem, there exists $A^{\star\star}\subseteq A'\setminus\{0\}$ with $|A^{\star\star}|\gtrsim N^{\frac{1}{2}-C\epsilon}$ such that

$$|A^{\star\star}\cdot A^{\star\star}|\lesssim N^{\frac{1}{2}+C\epsilon}$$

My question is why we have $A^{\star\star}=A''$. Do I understand it wrong?

Actually, I have tried to let the above $A^{\star\star}$ be a subset of $A’'\setminus\{0\}$ (by using the Balog-Szemerédi-Gowers Theorem) but it seems that we don’t have (or I cannot prove)

$$\left|\left\{(t,x_0,x_1)\in B\times A\times A’’:(1-t)x_0+tx_1\in A,t\neq0,1\right\}\right|\gtrsim N^{\frac{3}{2}-C\epsilon}$$

(The similar inequality (16) for $A’'$)

​​Could anyone help me understand the proof? Thanks for any help!

Answer 1

I think you're correct that (18) should read:

$$|A'' + A''| \lesssim N^{1/2+C\epsilon}.$$

The display before (16) tells you that for each $x_1 \in A' $, (and hence $x_1 \in A'' \subseteq A'$) one has:

$$|\{(t,x_0) \in B \times A : (1-t)x_0 + t x_1 \in A; t \neq 0, 1 \}| \gtrsim N^{1-C \epsilon}$$

Since $|A''|\gtrsim N^{1/2}$, after adjusting implicit constants, summing over elements in $A'$, gives us the estimate

$$|\{(t,x_0,x_1) \in B \times A \times A'' : (1-t)x_0 + t x_1 \in A; t \neq 0, 1 \}| \gtrsim N^{3/2-C \epsilon}$$

On the other hand, using (14) (e.g. $|A| \lesssim N^{1/2}$), the pigeonhole principle gives us (after adjusting implicit constants again) that for some $x_0$ (in fact the typical $x_0 \in A$) one has:

$$|\{(t,x_1) \in B \times A'' : (1-t)x_0 + t x_1 \in A; t \neq 0, 1 \}| \gtrsim N^{1-C \epsilon}.$$

This is the claim in the display following (18). One can then return to the argument as written.

Also let me point out that these arguments have been simplified and improved since the breakthrough paper of Bourgain-Katz-Tao. For instance, one can replace the repeated applications to BSG with a geometric argument. See, for instance, the paper of Stevens and de Zeeuw: https://arxiv.org/abs/1609.06284.