Nearly Sharp Uniform Projection Bounds and Superlinear Vitushkin Inequalities for Deterministic Self-Similar Sets
Abstract
For each α∈(0,1), we construct an explicit deterministic finite homogeneous planar iterated function system of similarities satisfying the strong separation condition, whose attractor Eα has positive finite one-dimensional Hausdorff measure and, for all sufficiently small δ>0, satisfies sup0≤θ<πH1(πθ(Nδ(Eα)))≲α(logδ−1)−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(Nδ(Eα))asympα(logδ−1)−1/2 as δ→0. Consequently, each Eα satisfies the quantitative Vitushkin comparison Fav(Nδ(Eα))≲αγ(Nδ(Eα))2α for every δ∈(0,1). These comparisons are superlinear for α>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.
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BibTeX
@misc{marshall2026nearly,
title = {{Nearly Sharp Uniform Projection Bounds and Superlinear Vitushkin Inequalities for Deterministic Self-Similar Sets}},
author = {Caleb Marshall},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.22908},
eprint = {2610.22908},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.22908},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Nearly Sharp Uniform Projection Bounds and Superlinear Vitushkin Inequalities for Deterministic Self-Similar Sets
AU - Caleb Marshall
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2610.22908
UR - https://arxiv.org/abs/2610.22908
AB - For each α∈(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>0, satisfies sup_0≤θ<πH^1(π_θ(N_δ(E_α)))≲_α(logδ^−1)^−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^−1)^−1/2 as δ→0. Consequently, each E_α satisfies the quantitative Vitushkin comparison Fav(N_δ(E_α))≲_αγ(N_δ(E_α))^2α for every δ∈(0,1). These comparisons are superlinear for α>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.
ER -