Endpoint inverse-logarithmic Favard decay for a finite self-similar system

Determine whether there exists an explicitly specified finite system of planar similarities satisfying the disk separation assumptions, with a noncollinear attractor E, such that Favard length obeys Fav(#1{δ}{E}) ≲ (log δ^{-1})^{-1} for every δ ∈ (0,1), and determine whether E can additionally satisfy the direction-uniform estimate sup_{0≤θ<π} ¹(π_θ(#1{δ}{E})) ≲ (log δ^{-1})^{-1} for every δ ∈ (0,1).

Background

The paper constructs explicitly specified finite homogeneous self-similar systems whose Favard lengths decay at every logarithmic power strictly below one, but it does not reach the endpoint exponent one. For the systems considered, the paper also establishes a universal lower bound of order (log δ{-1}){-1}, making this endpoint the fastest decay compatible with the available lower bound.

The question asks first for an existence result at the averaged Favard-length level and then for the stronger estimate that is uniform over every projection direction. The first formulation is identified as an existence version of Question 7.3 in Peres and Solomyak (2002).

References

Does there exist an explicitly specified finite system satisfying Definition~\ref{def:disk-separation}, with a noncollinear attractor $E$, such that

\Fav(#1{\delta}{E})\lesssim(\log\delta{-1}){-1}, \qquad \forall \delta \in (0,1)?

Can $E$ be chosen to satisfy the stronger estimate

\sup_{0\le\theta<\pi}1\bigl(\pi_\theta(#1{\delta}{E})\bigr) \lesssim(\log\delta{-1}){-1}\qquad\forall\delta\in(0,1)?

— Nearly Sharp Uniform Projection Bounds and Superlinear Vitushkin Inequalities for Deterministic Self-Similar Sets  (2610.22908 - Marshall, 16 Oct 2026) in Section 7, subsection “Endpoint restrictions,” Question