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

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 a