Furstenberg’s Times 2, Times 3 Conjecture (a Short Survey)

The uploaded survey states Rudolph’s theorem precisely and gives several correct proofs or proof sketches, including a clean derivation of Rudolph from Host’s theorem via ergodic decomposition and averaging q^k-pushforwards of the ×p-ergodic components with positive entropy (see the paragraph “Before presenting the proof, let us see why Host’s theorem implies Rudolph’s theorem” in Section 5) , and records the entropy proportionality (Proposition 3.1) ensuring positivity for all generators once one has it for some r ∈ ⟨p,q⟩ . By contrast, the model’s Step 2 incorrectly asserts that pushing ×p-ergodic components forward by ×q preserves their ×p-entropy, and then concludes that almost every ×p-ergodic component has positive ×p-entropy. Entropy is non-increasing under factor maps, so this step is not justified. The model consequently relies on pointwise equidistribution for μ-a.e. x, which it has not established, instead of the measure-level averaging argument used in the paper. Therefore the paper’s argument is correct, while the model’s proof contains a substantive gap.