EDIT: I moved this question to overflow forum, wasn't aware of the difference, sorry.
https://mathoverflow.net/questions/252396/inner-product-over-finite-fields
Let $F$ be a finite field,
For every $c \in F$, let $X_1, X_2,..., X_9, Y_1,..., Y_9$ be independent non-zero random variables over $F$.
Denote $X=(X_1,...,X_9)$, $Y=(Y_1,...,Y_9)$, also let $<X,Y>=\sum_{i}X_i Y_i$.
Question: show that $\sum_{x,y: <x,y>=c} (P( (X,Y)=(x,y) ))^{17/18} \leq 1$.
Remarks:
-feel free to swap $17/18$ for any other constant smaller then 1.
-I can prove this for flat distributions
-I can prove it for dimension 2, that is for $X_1, S_1$ instead of 18 random variables.
I'd be super grateful for the proof or sketch or idea that actually works :-).
Best regards,
Maciej