Furstenberg's $\times 2,\times 3$ orbit-dimension conjecture: must $\dim_H\overline{\{2^nx \bmod 1\}}+\dim_H\overline{\{3^nx \bmod 1\}}\ge 1$ hold for every irrational $x$?

A conjecture of H. Furstenberg (1970) predicts that for every irrational $x\in[0,1)$ the Hausdorff dimensions of the orbit closures of $x$ under the two maps $x\mapsto 2x \bmod 1$ and $x\mapsto 3x \bmod 1$ add up to at least $1$: multiplicative independence of the bases 2 and 3 should force any real number to have high orbit complexity in at least one base. The inequality is trivial for normal numbers, and Shmerkin and Wu proved in 2019, via their independent proofs of Furstenberg's intersection conjecture, that the set of potential exceptions has Hausdorff dimension zero; Chang, Li and Wu (2024, published 2026) determined the Hausdorff dimensions of the level sets of the dimension-sum function. Nevertheless the statement for every irrational point remains open and is described in the recent literature as far beyond current methods. The problem is to prove the inequality for all irrational $x$ (for the pair $(2,3)$, or more generally for every multiplicatively independent pair of integer bases $p,q\ge 2$), or to refute it by exhibiting an irrational $x$ whose two orbit-closure dimensions sum to strictly less than $1$.

Background

For an integer base $p\ge 2$, let $T_p:[0,1)\to[0,1)$ be the map $T_p(x)=px \bmod 1$, which shifts the base-$p$ digit expansion of $x$ one place to the left. The orbit of $x$ is the set $\{T_p^n x:n\ge 1\}$, and its closure records the asymptotic complexity of the base-$p$ expansion of $x$. Two integers $p,q\ge 2$ are called multiplicatively independent if $p^a\ne q^b$ for all positive integers $a,b$, equivalently $\log p/\log q\notin\mathbb{Q}$; the pair $(2,3)$ is the emblematic example. Hausdorff dimension, written $\dim_H$, is the standard fractal dimension of a subset of a metric space; for a compact set invariant under $T_p$ it equals the topological entropy of the associated symbolic dynamical system (the exponential growth rate of the number of admissible base-$p$ digit blocks, when points are coded by their digit sequences) divided by $\log p$. A number normal to base $p$ (one whose base-$p$ digit blocks all occur with their natural frequencies) has a dense $T_p$-orbit, so its orbit closure is the whole interval and has Hausdorff dimension $1$.

In his 1967 and 1970 work on rigidity in $\times p$ dynamics, Furstenberg posed a family of statements expressing that multiplicatively independent expansions of the same real number cannot both be simple. He settled the first two in the 1967 paper: the semigroup orbit $\{2^m3^n x \bmod 1\}$ of every irrational $x$ is dense, and the only closed subsets of $[0,1]$ invariant under both $T_2$ and $T_3$ are finite sets and the whole interval. The remaining members are conjectural: the only non-atomic Borel measure invariant and ergodic under both $T_2$ and $T_3$ is Lebesgue measure (the positive-entropy case of this measure rigidity is a theorem); and, in the 1970 paper 'Intersections of Cantor sets and transversality of semigroups', the intersection conjecture: if $A$ is closed and $T_p$-invariant, $B$ is closed and $T_q$-invariant, and $p,q$ are multiplicatively independent, then $\dim_H(A\cap(u+vB))\le\max\{0,\dim_H A+\dim_H B-1\}$ for all real $u,v$. Substantial parts of this program are settled: the sumset conjecture on sums of $\times 2$- and $\times 3$-invariant sets (Hochman and Shmerkin, 2012) and the intersection conjecture (Shmerkin, 2019, and Wu, 2019, independently; a third proof by Austin, 2022).

The present problem is the quantitative orbit-dimension conjecture, recorded as Conjecture 1.2 in Wu's 2019 paper and attributed to the same 1970 paper of Furstenberg: for multiplicatively independent $p,q$ and every irrational $x$, $\dim_H\overline{\{p^nx \bmod 1\}}+\dim_H\overline{\{q^nx \bmod 1\}}\ge 1$. It strengthens the intersection conjecture: if $A$ is $T_p$-invariant, $B$ is $T_q$-invariant and $\dim_H A+\dim_H B<1$, the intersection conjecture only forces $\dim_H(A\cap B)=0$, while the orbit-dimension conjecture predicts that $A\cap B$ can contain no irrational point at all, since the $T_p$- and $T_q$-orbit closures of such a point would be trapped inside $A$ and $B$ with dimensions summing below $1$. Wang (2014) restates the conjecture and constructs explicit non-normal numbers (points outside the full-measure set of normal points) for which both orbits are dense, so the inequality holds with both terms equal to $1$; this witnesses the conjecture on a set of full Hausdorff dimension but does not decide it for every irrational.

Problem Statement

Let $p,q\ge 2$ be multiplicatively independent integers, with $(p,q)=(2,3)$ as the case stated in Wang (2014). Furstenberg's orbit-dimension conjecture asserts that for every irrational $x\in[0,1)$, $\dim_H\overline{\{p^nx \bmod 1:n\ge 1\}}+\dim_H\overline{\{q^nx \bmod 1:n\ge 1\}}\ge 1.$ Decide the conjecture: either prove that the inequality holds for every irrational $x$ (for the pair $(2,3)$, or more generally for every multiplicatively independent pair $p,q$), or refute it by exhibiting an irrational $x\in[0,1)$ for which the sum of the Hausdorff dimensions of its two orbit closures is strictly less than $1$. The conjecture is known for Lebesgue-almost every $x$, and the set of exceptions is known to have Hausdorff dimension zero; the open content is precisely the statement for every irrational point. The broader descriptive program of classifying all joint behaviors of $\times 2$- and $\times 3$-orbits of a single point, or which non-generic digit patterns in the two bases can co-occur, contains this conjecture as its central quantitative open problem; that program as a whole is not a single well-posed question and is not part of what an answer must settle.

References

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