TITLE: Is the exponent $2$ sharp in the Balog-Szemerédi-Gowers Theorem?
QUESTION [6 upvotes]: The Balog-Szemerédi-Gowers theorem can be stated in the following form: let $A,B$ be subsets of $\mathbb{Z}/n\mathbb{Z}$ (say) with equal cardinality, such that
$$
\|1_A*1_B\|_2 \ge K^{-1} \|1_A\|_1 \|1_B\|_2. 
$$
Then there exist $A'\subset A$, $B'\subset B$ with $|A'|\ge K^{-C}|A|$, $|B'|\ge K^{-C}|B|$ such that $|A'+B'|\le K^C |B'|$.
Here $C$ is an absolute constant.
My questions is: what happens if the $2$-norm is replaced by a $q$-norm? If $q\in (1,2)$, then it follows from Hölder that
$$
\|1_A*1_B\|_q \ge K^{-1} \|1_A\|_1 \|1_B\|_q 
$$
implies
$$
\|1_A*1_B\|_2 \ge K^{-\frac{q}{2(q-1)}} \|1_A\|_1 \|1_B\|_2,
$$
so the conclusion still holds, with a constant $C$ that depends on $q$ (blowing up as $q\downarrow 1$).
What about $q>2$? More precisely:

Is there any $q>2$ such that the conclusion of the B-S-G Theorem continues to hold if one assumes that $\|1_A*1_B\|_q \ge K^{-1} \|1_A\|_1 \|1_B\|_q $ (possibly with a constant $C$ that depends on $q$)?

REPLY [2 votes]: Sorry to answer my own question, but I realized the answer is easy, and similar to the case $q\in (1,2)$: yes, the B-S-G theorem holds for any $q\in (2,+\infty)$: if $q>2$,
$$
\|1_A*1_B\|_q^q \le \|1_A*1_B\|_2^2 \|1_A*1_B\|_\infty^{q-2} \le \|1_A*1_B\|_2^2 |A|^{q-2},
$$
so if $\|1_A*1_B\|_q \ge K^{-1} \|1_A\|_1 \|1_B\|_q$, it follows that
$$
|A|^q |B| K^{-q} \le  \|1_A*1_B\|_q^q \le \|1_A*1_B\|_2^2 |A|^{q-2},
$$
and then
$$
\|1_A*1_B\|_2 \ge K^{-q/2} \|1_A\|_1 \|1_B\|_2,
$$
and we can apply the Balog-Szemerédi-Gowers Theorem.
This is simpler than I expected, but since the question got a few upvotes it might be useful to someone.