The Furstenberg’s conjecture on dimension of sum of invariant sets in true

Analysis 1 Minute

You need to know: Closed subset of [0,1], multiplication of a set by a constant s\cdot A = \{s\cdot a: a \in A\}, sum of two sets A + B = \{a + b : a \in A; b \in B\}, Hausdorff dimension \text{dim}(A) of set A \subset [0,1].

Background: For real number x, let \left \lfloor{x}\right \rfloor be the largest integer not exceeding x, and let \text{frac}(x)=x-\left \lfloor{x}\right \rfloor. For integer m, let T_m:[0,1)\to[0,1) be a function given by T_m(x)=\text{frac}(mx). We say that set S \subset [0,1] is invariant under T_m if T_m(x) \in S for every x \in S.

The Theorem: On 11th October 2009, Michael Hochman and Pablo Shmerkin submitted to arxiv a paper in which they proved the following result. Let X; Y \subseteq [0,1] be closed sets which are invariant under T_2 and T_3, respectively. Then, for any s \neq 0, \text{dim}(X + s\cdot Y ) = \min\{1, \text{dim} X + \text{dim} Y\}.

Short context: The Theorem confirms a long-standing conjecture of Furstenberg made in late 1960th. See here for a theorem resolving another related conjecture of Furstenberg in a later work.

Links: Free arxiv version of the original paper is here, journal version is here.

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Published by Bogdan Grechuk

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