Assouad Dimensions and Spectra of Graphs of Weierstrass-Type Functions with Rapidly Growing Frequencies
Abstract
We study the Assouad dimension, lower dimension, and Assouad spectrum of graphs of the functions f(x)=∑n=1∞anϕ(bnx+θn), where ϕ is a nonconstant 1-periodic C2 function, an,bn>0, (θn) is an arbitrary real sequence, ∑n=1∞an<∞, and bn+1/bn→∞. We give sufficient conditions for the graph over every nondegenerate compact interval to have Assouad dimension two or lower dimension one. For an=bn−α, 0<α<1, an explicit frequency-gap condition ensures that the Assouad spectrum equals two for α≤ϑ<1. Writing β*=limsupn→∞log bn+1/log bn, we show that, if β*>(2−α)/α, the spectrum attains the general upper bound for α-Hölder graphs whenever (2−α)/β*≤ϑ<α and ϑ>0. When β*=∞, this determines the complete spectrum. Together with Barański's formulas, our results yield graphs whose lower, Hausdorff, upper box, and Assouad dimensions are four distinct numbers.
Related
- Assouad type dimensions of generalized affine fractal interpolation functions and their applications
- On the Assouad dimension of Weierstrass function graphs
- Explicit Robin Green's Functions and Resonance Spectra for the Helmholtz Equation on Balls in All Dimensions
- Algebraic independence of the exponential and Weierstrass ℘-functions
- Integer points for iterated Weierstrass elliptic functions
BibTeX
@misc{xiong2026assouad,
title = {{Assouad Dimensions and Spectra of Graphs of Weierstrass-Type Functions with Rapidly Growing Frequencies}},
author = {Dong-Hong Xiong and Jun Jason Luo},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.37388},
eprint = {2609.37388},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.37388},
}
RIS
TY - UNPB
TI - Assouad Dimensions and Spectra of Graphs of Weierstrass-Type Functions with Rapidly Growing Frequencies
AU - Dong-Hong Xiong
AU - Jun Jason Luo
PY - 2026
DA - 2026/10/17
PB - arXiv
DO - 10.48550/arxiv.2609.37388
UR - https://arxiv.org/abs/2609.37388
AB - We study the Assouad dimension, lower dimension, and Assouad spectrum of graphs of the functions f(x)=∑_n=1^∞a_nϕ(b_nx+θ_n), where ϕ is a nonconstant 1-periodic C^2 function, a_n,b_n>0, (θ_n) is an arbitrary real sequence, ∑_n=1^∞a_n<∞, and b_n+1/b_n→∞. We give sufficient conditions for the graph over every nondegenerate compact interval to have Assouad dimension two or lower dimension one. For a_n=b_n^−α, 0<α<1, an explicit frequency-gap condition ensures that the Assouad spectrum equals two for α≤ϑ<1. Writing β_*=limsup_n→∞log b_n+1/log b_n, we show that, if β_*>(2−α)/α, the spectrum attains the general upper bound for α-Hölder graphs whenever (2−α)/β_*≤ϑ<α and ϑ>0. When β_*=∞, this determines the complete spectrum. Together with Barański's formulas, our results yield graphs whose lower, Hausdorff, upper box, and Assouad dimensions are four distinct numbers.
ER -