Furstenberg's Conjecture on ×2 and ×3 Invariant Sets

Classify closed subsets T ⊂ [0,1] such that T = 2T mod 1 and T = 3T mod 1. Conjecture: only {0}, {1}, and the whole interval.

Grew out of Furstenberg's 1967 disjointness theorem for ×2 and ×3 (times-two and times-three have no common non-trivial closed invariant sets beyond finite orbits of rationals). Rudolph and Johnson (1990-91) proved the measure-theoretic disjointness under positive entropy; Host, Lindenstrauss–Peres–Schlag and others extended rigidity to fractal dimensions. The pure topological classification above is where the wall sits.