PSU Mark
Eberly College of Science Mathematics Department

Meeting Details

For more information about this meeting, contact Stephen Simpson.

Title:Sumsets, fractals and dynamics
Seminar:Department of Mathematics Colloquium
Speaker:Michael Hochman, Princeton University
Abstract:
If A,B are sets in the line of Hausdorff dimension less than one, then their sumset {a+b : a in A, b in B} can have dimension anywhere between max{dim(A),dim(B)} and min{1,dim(A)+dim(B)}. The valdity of these bounds is elementary and any intermediate value is possible, but for "typical" pairs of sets the upper bound is the correct dimension; i.e., the sumset is as large as it can be. This statement is valid under several natural interpretations of "typical". It is however much harder to establish this behavior for even some of the simplest concrete fractal sets. In this direction, Furstenberg made a number of conjectures in the late 1960's, predicting that sets which arise from "arithmetically independent" dynamics should behave in this sense and others as "typical" pairs of sets. In particular, this should happen when A,B are subsets of [0,1] which are invariant, respectively, under 2x mod 1 and 3x mod 1. I will describe joint work with Pablo Shmerkin in which we prove this conjecture and some related statements about projections of fractal sets and measures which arise in a dynamical context.

Room Reservation Information

Room Number:MB114
Date:03 / 18 / 2010
Time:04:00pm - 05:00pm