
Speaker Michael Hochman Host Yuval Peres Affiliation Veblen Instructor at Princeton University Duration 00:58:56 Date recorded 25 January 2010 By a classical theorem of Marstrand, if X is a set in the plane then almost every linear projection of X onto a line has the largest possible dimension, i.e. min1,dim(X). However, in general the dependence of this number on the projection is quite bad and is not well understood even in very simple examples. In joint work with Pablo Shmerkin we show that for a large class of fractals which arise from arithmetic, dynamical or combinatorial constructions there is some semicontinuity in the projection. Using this we establish a number of results in fractal geometry. I will focus mainly on the proof of the following conjecture of Furstenberg from the early 1970s: If A,B are closed subsets of [0,1] which are, respectively, invariant under times2 and times3 (mod 1), then for every nonzero t, the dimension of the sumset is "as large as exspected", i.e. dim(A + tB) = min1 , dim(A) + dim(B)
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