Hausdorff measure positivity for projections in the intermediate parameter range

Determine for which parameters d with 1/9≤d≤1/6 the four-corner Cantor set C(d) satisfies H^{s_d}(p_θ(C(d)))>0 for Lebesgue-almost-every direction θ∈[0,π), where s_d=log 4/log(1/d) and p_θ is the orthogonal projection onto direction θ.

Background

For the four-corner Cantor set C(d)=K_d×K_d, the paper studies whether the s_d-dimensional Hausdorff measure of its orthogonal projection is positive for almost every direction. The cited prior results established the positive conclusion for d<1/9 and showed that it fails for 1/6<d<1/4.

The paper proves failure of positivity for the subrange δ<d≤1/6, where δ≈0.1551249839 is the specified zero of a polynomial. Thus, the explicitly identified intermediate range was historically unresolved, and the paper provides only a partial resolution of it.