The high-dimensional Weierstrass functions

2024/04/10 by Haojie Ren, Ren, Haojie, Weixiao Shen +1
Mathematics · #Advanced Banach Space Theory #Advanced Topology and Set Theory #Classical Analysis and ODEs (math.CA) #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2404.06778

openalex publication_date 2024/04/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For a real analytic periodic function ϕ:ℝ→ℝ^d, an integer b ≥ 2 and λ∈(1/b,1), we prove that the box dimension and the Hausdorff dimension of the graph of the Weierstrass function W(x)=∑_n=0^∞λⁿϕ(bⁿx) are both equal to min\log_λ^-1b, 1+( d-q )(1+log_bλ)\, where q = q(ϕ, b, λ) denotes the maximum dimension of all linear spaces V < ℝ^d such that the projection π_V W is Lipschitz.

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