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 positive constant smaller then $1$.

-I can prove this for flat distributions (using min-entropy).

-I can prove it for dimension $2$ (with constant $1/2$ instead of $17/18$), that is for $X_1, Y_1$ instead of $18$ random variables.

I'd be super grateful for the proof or sketch or idea that actually works :-).

Best regards,

Maciej

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 $\langle X,Y \rangle =\sum_{i}X_i Y_i$.

Question: Show that $$\sum_{x,y: \langle x,y\rangle =c} (P[(X,Y)=(x,y) ])^{17/18} \leq 1.$$

Remarks:

-feel free to swap $17/18$ for any other positive constant smaller then $1$.

-I can prove this for flat distributions (using min-entropy).

-I can prove it for dimension $2$ (with constant $1/2$ instead of $17/18$), that is for $X_1, Y_1$ instead of $18$ random variables.

I'd be super grateful for the proof or sketch or idea that actually works :-).

Best regards,

Maciej

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 positive constant smaller then $1$.

-I can prove this for flat distributions (using min-entropy).

-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

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 positive constant smaller then $1$.

-I can prove this for flat distributions (using min-entropy).

-I can prove it for dimension $2$ (with constant $1/2$ instead of $17/18$), that is for $X_1, Y_1$ instead of $18$ random variables.

I'd be super grateful for the proof or sketch or idea that actually works :-).

Best regards,

Maciej