Furstenberg's ×2×3 Conjecture

Claim: if x is irrational, the set { \(2^{m}\) \(3^{n}\) x mod 1 : m, n ∈ ℕ } is dense in [0, 1]. Equivalently, the action of the semigroup generated by ×2 and ×3 on the circle is minimal on every infinite orbit closure.

Born from Furstenberg's 1967 disjointness paper, where he proved the topological disjointness of ×2 and ×3 (no common non-trivial invariant closed set). The density conjecture is the simplest question about their JOINT action, and it has become the model problem for rigidity of ℤ²-actions. Rudolph (1990) and Johnson proved the measure-theoretic analogue under a positive-entropy assumption; Host and others extended the framework. Without entropy assumptions, even the statement above resists every mixing technology available.