Furstenberg’s zero-dimensional exceptional case

Determine whether Furstenberg’s conjectured dimension inequality holds for every irrational real number when one of the two orbit-closure dimensions is zero, thereby resolving the case that remains open for multiplicatively independent integer bases.

Background

Furstenberg’s conjecture concerns the Hausdorff dimensions of the orbit closures generated by multiplication by two multiplicatively independent integers in different bases. The paper notes that Shmerkin and Wu proved that the set of exceptions has Hausdorff dimension zero, but identifies the case in which one of the dimensions is zero as the principal unresolved situation.