Geometric and algebraic methods in Erdos type problems

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The main objective of the proposed research program is to develop new and improved techniques to attack arithmetic problems in discrete geometry and additive combinatorics.Several central problems in different parts of mathematics can be translated into questions in discrete geometry. In many cases, such connections were discovered relatively recently due to pioneering works of great mathematicians like Jean Bourgain, Tim Gowers, and Terry Tao. As it turned out there are classical problems in discrete geometry which have direct impacts in harmonic analysis, combinatorics, number theory, and theoretical computer science. This field even has its unique Mathematics Subject Classification code: 52C10, Erdos problems and related topics of discrete geometry. Many of these problems can be formulated as counting (bounding) the maximum number of incidences between lines/curves/planes/surfaces and points. (We say that a point is incident to a line (or curve or surface) if the point lies on the line.) Incidence bounds provide inside information about the arithmetic-geometric structure of the underlying field. A well known example - and an important part of my research - is the sum-product problem: given a finite set of integers, A, is it possible that both the sumset, A+A, and the product set, A*A, are small? (The sumset and product set are the set of pairwise sums and products of elements from A.) For example if A is the set of the first n natural numbers then the sumset is small, it has cardinality 2n-1, while the product set is almost quadratic in n=|A|. If A is a geometric progression then the product set is small, but then the sumset is quadratic in |A|. Erdos and Szemeredi is conjectured that |A+A|+|A*A|>|A|2-epsilon, where epsilon goes to zero as |A| goes to infinity. All significant improvements in this problem have come from (discrete) geometry, by understanding the connections between the geometry of the plane and the arithmetic of the underlying field. The particular problem I will focus on is Erdos' Unit Distances Problem: What is the maximum number of unit distances among pairs of n points on the plane? Erdos conjectured that the upper bound on unit distances is n1+epsilon, where epsilon goes to zero as n goes to infinity. This is a 70 year old problem in which the best upper bound, n4/3, was given more than 30 years ago. I plan to improve this upper bound. There are examples of metrics similar to the Euclidean, where the number of unit distances is n4/3 , so any possible improvement should use more than the combinatorics of unit circle arrangements. Improving this bound might seem to be overly ambitious, but recent developments in using algebraic methods to tackle similar problems make the plan look more feasible.