Classify the thin side for multiplicatively dependent bases

Classify the thin-side behavior of the radial projection problem for multiplicatively dependent bases when the initial-block thickness parameters satisfy c_a+c_b<1.

Background

The paper establishes an exact all-observers interior threshold c_a+c_b=1 only within the multiplicatively independent initial-block family. Below this threshold, the paper constructs bad observers when the bases are multiplicatively independent, including open regions and dense arithmetic families. The corresponding thin-side behavior for multiplicatively dependent bases is not resolved by the results presented.

References

The condition c_a+c_b=1 is an exact all-observers boundary only within the multiplicatively independent initial-block family. The thin side for multiplicatively dependent bases is not classified.

— A thickness boundary and modular obstructions for two-set radial projections  (2609.03400 - Miwa, 3 Sep 2026) in Section Discussion, first bullet