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

For each $\alpha \in (0,1)$, we construct an explicit deterministic finite homogeneous planar iterated function system of similarities satisfying the strong separation condition, whose attractor $E_{\alpha }$ has positive finite one-dimensional Hausdorff measure and, for all sufficiently small $\delta >0$, satisfies ${\sup }_{0\le \theta <\pi }{H}^1({\pi }_{\theta }(N_{\delta }(E_{\alpha }))){\lesssim }_{\alpha }(\log {\delta }^{-1}{)}^{-\alpha }$. The defining system depends only on $\alpha$ and is fixed at every scale. For these same sets, we prove the sharp analytic capacity law $\gamma (N_{\delta }(E_{\alpha })){\asymp }_{\alpha }(\log {\delta }^{-1}{)}^{-1\mathrm{/}2}$ as $\delta \to 0$. Consequently, each $E_{\alpha }$ satisfies the quantitative Vitushkin comparison $\mathrm{Fav}(N_{\delta }(E_{\alpha })){\lesssim }_{\alpha }\gamma (N_{\delta }(E_{\alpha }){)}^{2\alpha }$ for every $\delta \in (0,1)$. These comparisons are superlinear for $\alpha >1\mathrm{/}2$, with exponents arbitrarily close to the optimal upper limit two for this class of self-similar sets. A completely specified sixteen-map system gives $\alpha =53\mathrm{/}100$ and comparison exponent $53\mathrm{/}50$. 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.