Seminar 4.9.15

Title: A discrete to continuous framework for projection theorems

Speaker: Daniel Glasscock

Abstract: Projection theorems for planar sets take the following form: the image of a “large” set A2 under “most” orthogonal projections πθ (to the line through the origin corresponding to θS1) is “large”; the set of those directions for which this does not hold is “small.” The first such projection theorem was given by J. M. Marstrand in 1954: assuming the Hausdorff dimension of A is less than 1, the Hausdorff dimension of πθA is equal to that of A for Lebesgue-almost every θS1.

Recent progress has been made on some fundamental problems in geometric measure theory by discretizing and using tools from additive combinatorics.  In 2003, for example, J. Bourgain building on work of N. Katz and T. Tao, used this approach to prove that a Borel subring of  cannot have Hausdorff dimension strictly between 0 and 1 (a result shown independently by G. A. Edgar and C. Miller), answering a question of P. Erd\H{o}s and B. Volkmann. The goal of my talk is to explain a discrete approach to continuous projection theorems.  I will show, for example, how Marstrand’s original theorem and a recent result of Bourgain and D. Oberlin can be obtained combinatorially through their discrete analogues. This discrete to continuous framework connects finitary combinatorial techniques to continuous ones and hints at further parallels between the two regimes.