# European Congress of Mathematics

Amsterdam, 14–18 July, 2008

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#### pp: 233–251

#### DOI: 10.4171/077-1/11

^{[1]}(1) School of Mathematics, Institute for Advanced Study, 1 Einstein Drive, NJ 08540, PRINCETON, UNITED STATES

This is a survey of a line of research in arithmetic combinatorics. It is centered around so-called sum-product phenomena in various settings and its applications to problems in number theory, computer science, spectral and ergodic theory. More speciﬁcally, the sum-product results in ﬁnite ﬁelds and residue rings lead to new bounds on exponential sums of various types and in fact provide the ﬁrst non-trivial estimates. A typical result in this spirit are bounds on Gauss sums for small multiplicative subgroups. Product theorems in matrix spaces derived from the scalar theory enable one to prove various conjectures on the expansion of Cayley graphs and the existence of spectral gaps for Hecke operators, most notably in SL_{2}(*q*) and SU(2). Those in turn lead to an extension of Selberg’s theorem for congruence subgroups and new results on prime sieving in non-elementary subgroups of SL_{2}(ℤ). Finally, Furstenberg’s “stiffness problem” for toral actions of subgroups of SL_{2}(ℤ) as well as a quantitative equidistibution property of the orbits are described.

*Keywords: *Sum-product, exponential sum, expander graph