Quantitative cusp control from Fefferman approximation theorems

Determine whether Fefferman's quantitative improvements of Whitney's approximation theorem provide quantitative control over the cusps of the simply connected cusp domain used in the construction separating Rubel_0(1) from Rubel_0(3), and determine whether those cusps can be asymptotic to two tangent circles.

Background

The construction proving Rubel_0(1) is not contained in Rubel_0(3) uses a real-analytic approximation to a smooth function on (0,1), followed by analytic continuation to a cusp domain. The paper notes that an explicit or quantitatively controlled approximant could yield more information about the geometry of the required cusps. It specifically asks whether Fefferman's quantitative versions of Whitney's theorem can establish such control and whether tangent-circle asymptotics are possible.