In this talk I will discuss a finitary version of projection theorems in fractal geometry. Roughly speaking, a projection theorem says that, given a subset in the Euclidean space, its orthogonal projection onto a subspace is large except for a small set of exceptional directions. There are several ways to quantify "large" and "small" in this statement. We will place ourself in a discretized setting where the size of a set is measured by its delta-covering number : the minimal number of balls of radius delta needed to cover the set, where delta > 0 is the scale. The pioneering work of Bourgain relates the problem to sum-product phenomenon in arithmetic combinatorics and proved a discretized projection theorem for projections onto lines. I will present an extension to Bourgain's result and its fractal geometric consequences.
Tue, 31/10/2017 - 14:00 to 15:00