Abstract. Let A be a subset of a finite field F: = Z/qZ for some prime q. If |F | δ < |A | < |F | 1−δ for some δ> 0, then we prove the estimate |A + A | + |A · A | ≥ c(δ)|A | 1+ε for some ε = ε(δ)> 0. This is a finite field analogue of a result of [ESz1983]. We then use this estimate to prove a Szemerédi-Trotter type theorem in finite fields, and obtain a new estimate for the Erdös distance problem in finite fields, as well as the three-dimensional Kakeya problem in finite fields. 1.

Given a set P of n points in convex position in the plane, we prove that there exists a point p 2 P such that the number of distinct distances from p is at least d(13n - 6)/36e. The best previous bound, dn=3e, from 1952, is due to Leo Moser.

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The aim of this course is to tour the highlights of arithmetic combinatorics- the combinatorial estimates relating to the sums, differences, and products of finite sets, or to related objects such as arithmetic progressions. The material here is of course mostly combinatorial, but we will also exploit the Fourier transform at times. We