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Dimension of Self-similar Measures and Additive Combinatorics

Wednesday 2nd July 2014 - 10:00 to 10:50
INI Seminar Room 1
I will discuss recent progress on the problem of computing the dimension of a self-similar set or measure in $\mathbb{R}$ in the presence of non-trivial overlaps. It is thought that unless the overlaps are "exact" (an essentially algebraic condition), the dimension achieves the trivial upper bound. I will present a weakened version of this that confirms the conjecture in some special cases. A key ingredient is a theorem in additive combinatorics that describes in a statistical sense the structure of measures whose convolution has roughly the same entropy at small scales as the original measure. As time permits, I will also discuss the situation in $\mathbb{R}^d$.
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University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons