The Furstenberg’s conjecture on the intersections of invariant sets is true

Analysis, Number theory 1 Minute

You need to know: Set {\mathbb Q} of rational numbers, 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 25th September 2016, Pablo Shmerkin submitted to arxiv a paper in which he proved the following result. Let p,q \geq 2 be positive integers such that \frac{\log p}{\log q}\not\in{\mathbb Q}. Let A; B \subseteq [0,1] be closed sets which are invariant under T_p and T_q, respectively. Then for all real numbers u and v, \text{dim}((u\cdot A + v) \cap B) \leq \max\{0, \text{dim}(A) + \text{dim}(B) - 1\}.

Short context: The Theorem confirms a long-standing conjecture of Furstenberg made in 1969. In fact, the Furstenberg’s conjecture corresponds to the case u=1 and v=0 of the Theorem. On 26th September 2016, Meng Wu submitted to arxiv a paper with independent and different proof of the same result. Also, see here for an earlier theorem resolving another related conjecture of Furstenberg.

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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