Local entropy averages and projections of fractal measures

Опубликовано 17 августа 2016, 20:34

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. min{1,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 semi-continuity 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 times-2 and times-3 (mod 1), then for every non-zero t, the dimension of the sumset is 'as large as exspected', i.e. dim(A + tB) = min{1 , dim(A) + dim(B)}