A thickness boundary and modular obstructions for two-set radial projections

Abstract: Let $K_{a,m}$ and $K_{b,n}$ be missing-digit Cantor sets with initial consecutive digit sets, and write $c_a=m/(a-1)$ and $c_b=n/(b-1)$. We prove that if $c_a+c_b\geq 1$, then the radial projection of $K_{a,m}\times K_{b,n}$ from every observer has nonempty interior; no multiplicative-independence assumption is needed for this implication. Conversely, when the bases are multiplicatively independent and $c_a+c_b&lt;1$, we exhibit explicit unbounded open sets of observers for which the radial image is compact and nowhere dense. Rational observers with the same property are dense in each exterior corner region and occur arbitrarily close to the four corners of the product. Thus $c_a+c_b=1$ is the exact threshold for the all-observers interior property within the multiplicatively independent initial-block family. In particular, this supplies an explicit two-set counterexample to the nonempty-interior conclusions of two conjectures of Yu. We also establish a sufficient modular phase obstruction for affine translates of such sets. Combining it with a fixed-pin positive-measure theorem of Banaji and Yu yields consecutive-block division sets that are compact, perfect, of positive Lebesgue measure, and nowhere dense, with both factor dimensions tending to one. For the same family, Fourier $l<sup>1$-dimension estimates and the incidence argument used in Yu's product theorem imply that the two self-products contain intervals for all sufficiently large $r$ and $s$, while the cross-division set remains nowhere dense.