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 |