Dimensions of non-resonant rotated Sierpiński-gasket intersections

Determine the Hausdorff and box dimensions of the intersection of the standard planar Sierpiński gasket with its rotation by every non-resonant angle, and prove or refute the conjectured equality that both dimensions equal 2H(S)-2.

Background

The paper studies the intersection Sθ=S∩RθS\mathcal S_\theta=S\cap \mathbf R_\theta S of the standard planar Sierpiński gasket SS with a copy rotated through angle θ\theta. For resonant angles satisfying eiθ∈Q(ω)e^{i\theta}\in\mathbb Q(\omega), the authors establish finite-type structure and compute both Hausdorff and box dimensions using the spectral radius of an explicit finite transition matrix.

For non-resonant angles, the paper proves only the almost-everywhere upper box-dimension bound dim⁡‾B(Sθ)≤2H(S)−2\overline{\dim}_{\mathrm B}(\mathcal S_\theta)\le 2H(S)-2. The matching Hausdorff lower bound remains unresolved, particularly because the two gaskets have fixed coincident centres rather than a variable generic translation. Numerical intersection-count experiments support the conjecture that, at every non-resonant angle, both the Hausdorff and box dimensions equal $2H(S)-2$.

References

The spectral-radius calculation displays the angular variation of the resonant dimensions. The non-resonant counting experiments support the following conjecture, whose lower bound remains open.

Let $\theta\in[0,\pi/3]$ with $e{i\theta}\notin\mathbb Q(\omega)$. Then

H(\mathcal S_\theta)=B(\mathcal S_\theta)=2H(S)-2.

— Moiré Pattern of Rotated Sierpiński Gaskets  (2609.30235 - Neroli, 24 Sep 2026) in Section 3.2, “Numerical experiments and the conjecture,” Conjecture \ref{conj:non-resonant-dimension}; see also Section 3, “The lower bound and numerical evidence”