Nearly Sharp Uniform Projection Bounds and Superlinear Vitushkin Inequalities for Deterministic Self-Similar Sets

Abstract: For each $α\in(0,1)$, we construct an explicit deterministic finite homogeneous planar iterated function system of similarities satisfying the strong separation condition, whose attractor $E_α$ has positive finite one-dimensional Hausdorff measure and, for all sufficiently small $δ&gt;0$, satisfies $\sup_{0\leθ&lt;π}H<sup>1(π<em>θ(N</em>δ(E_α)))\lesssim_α(\logδ<sup>{-1})<sup>{-α}$. The defining system depends only on $α$ and is fixed at every scale. For these same sets, we prove the sharp analytic capacity law $γ(N_δ(E_α))\asymp_α(\logδ<sup>{-1})<sup>{-1/2}$ as $δ\rightarrow0$. Consequently, each $E_α$ satisfies the quantitative Vitushkin comparison $$\operatorname{Fav}(N_δ(E_α))\lesssim_αγ(N_δ(E_α))<sup>{2α}$$ for every $δ\in(0,1)$. These comparisons are superlinear for $α&gt;1/2$, with exponents arbitrarily close to the optimal upper limit two for this class of self-similar sets. A completely specified sixteen-map system gives $α=53/100$ and comparison exponent $53/50$. Inspired by Falconer's Venetian blind constructions, the projection argument combines exact collinear cylinder subfamilies, quantitative distribution of word orientations, and a cover indexed by first successful symbolic blocks. The rotation parameters are explicit powers of a real algebraic number, linking the construction to simultaneous Diophantine approximation. The capacity estimate follows from the curvature characterization of analytic capacity and a telescoping inequality for cylinder masses.