2025/07/21 by Zoltán Buczolich, Antti Käenmäki, Buczolich, Zoltán +3
Mathematics · #26A16 #28A50 #46E35 #60G17 #Advanced Banach Space Theory #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical Dynamics and Fractals #Point processes and geometric inequalities #Primary 28A78 #Secondary 26E15
paper · pdf · doi:10.48550/arxiv.2507.15591
openalex publication_date 2025/07/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The α-Weierstrass function is defined as Wgα,b(x) = ∑k=0∞ b-αk g(bk x), where g is a Lipschitz function on the unit circle. For a prevalent α-Weierstrass function, we prove that the upper Minkowski dimension of every level set is at most 1-α, and the Hausdorff dimension of almost every level set equals 1-α with respect to its occupation measure. We further demonstrate that the occupation measure of a prevalent α-Weierstrass function is absolutely continuous with respect to the Lebesgue measure. Consequently, the result on the Hausdorff dimension of level sets applies to a set of level sets with positive Lebesgue measure. A central tool in our analysis is the Weierstrass embedding. For a sufficiently large dimension d, we construct Lipschitz functions g0, g1, …, gd-1 such that the mapping x ↦ (Wg0α,b(x), Wg1α,b(x), …, W_gd-1α,b(x)) is α-bi-Hölder. We also prove that such an embedding requires at least 1/α coordinate functions.