Sum-product estimates via directed expanders

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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Published 1 January 2008