Sum-product estimates via directed expanders

Van Vu
Rutgers University

Let $\F_q$ be a finite field of order $q$
and $P$ be a polynomial in $\F_q[x_1, x_2]$. For a set $A \subset
\F_q$, define $P(A):=\{P(x_1, x_2) | x_i \in A \}$. Using certain
constructions of expanders, we characterize all polynomials $P$ for
which the following holds \vskip2mm \centerline{\it If $|A+A|$ is small (compared to $|A|$),then $|P(A)|$ is large.} \vskip2mm \noindent The case $P=x_1x_2$ corresponds to the well-known sum-product problem.

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