Assouad and Lower Dimensions of Graphs of Weierstrass-Type Functions with Rapidly Growing Frequencies

Jun Jason Luo

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arXiv · 2609.37388

Assouad and Lower Dimensions of Graphs of Weierstrass-Type Functions with Rapidly Growing Frequencies

Jun Jason Luo

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Abstract

We study the Assouad and lower dimensions of graphs of Weierstrass-type functions of the form \[ f(x)=\sum_{n=1}^{\infty}a_nϕ(b_nx+θ_n), \qquad x\in\mathbb R, \] where $ϕ$ is a nonconstant $C^2$ function of period one, $a_n,b_n>0$, $(θ_n)$ is an arbitrary sequence of real numbers, $\sum_{n=1}^{\infty}a_n<\infty$, and $b_{n+1}/b_n\to\infty$. We give quantitative conditions under which the graph over every nondegenerate compact interval has Assouad dimension two or lower dimension one. In particular, when $a_n=b_n^{-α}$, we obtain explicit frequency-gap conditions under which the Assouad spectrum of the graph is equal to $2$ for $α\leq\vartheta<1$. When the logarithmic frequency ratios converge, we also determine the spectrum on a nonempty interval below $α$. Together with Barański's formulas for the Hausdorff and box dimensions, these results yield graphs whose lower, Hausdorff, upper box, and Assouad dimensions are four distinct numbers.

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BibTeX RIS

Jun Jason Luo. 2026-09-29. Assouad and Lower Dimensions of Graphs of Weierstrass-Type Functions with Rapidly Growing Frequencies. https://arxiv.org/abs/2609.37388

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