Increase of entropy under convolution and self-similar sets with overlaps

Abstract. Sets that consist of finitely many smaller-scale copies of itself are known as self-similar. Due to the likely irregularity in their structure, the size of these sets is often measured in the form of dimension. The existence of tools that can be used to calculate this quantity depends greatly on whether the cylinders of which the set consists of are sufficiently separated from each other. If this is the case, the dimension of the set is known to equal its similarity dimension, a quantity that is relatively easy to calculate. There is a long-standing open conjecture stating that, for a general set on the real line, the only case in which the dimension of the set does not equal its similarity dimension, is when at some scale there is an exact overlap among the cylinders of the set. The main result in this thesis is a step towards showing that this is indeed the case; in the presence of an exact overlap, the distance between the cylinders of the set decreases exponentially. This result is due to M. Hochman and it appeared in his paper “On self-similar sets with overlaps and inverse theorems for entropy” (2012) and forms the basis of our discussion in Section 4. In Section 1, we analyse the growth of entropy of a probability measure under convolution. The main result of this section is a generalization of the Freiman theorem from additive combinatorics to the fractal regime, stating that if the entropy of a convolution measure is not too large, then one of the marginal measures has to be either locally uniform or locally atomic. This result is also due to Hochman and is one of the main tools used in proving the results of Section 4. In Sections 2 and 3, we introduce the concepts of dimension and the main tools required in understanding the structure of a self-similar set or measure with sufficient separation conditions in place. Most of the results here can be found in any text-book concerning fractal geometry, e.g. Falconer’s “Fractal Geometry” (1990)