Assouad type dimensions of generalized affine fractal interpolation functions and their applications
Abstract
In this article, we investigate the Assouad spectrum and Assouad dimension of graphs of fractal functions generated by generalized affine iterated function systems. We establish upper and lower bounds for the Assouad spectrum in terms of the scaling functions and the underlying partition of the generalized affine construction. If the scaling function is Lipschitz continuous and the partition is uniform, we obtain an explicit expression for the Assouad spectrum of the associated graph. These results provide a connection between the parameters defining the generalized affine fractal function and the local multiscale geometry of its graph. As applications, we consider two classic examples of generalized affine fractal functions, namely the Weierstrass and Takagi functions. For the classical Weierstrass function W, whose graph ΓW has the box dimension 2+logNλ, we obtain dimAθ(ΓW) ≤ (2+logNλ−θ)/(1−θ), θ∈ (0,logN1/λ). and if λ2 N <1, we get dimA(ΓW)≥ 1 +logN(1/λ). For the classical Takagi function T with graph ΓT, we show that dimAθ(ΓT)=1, θ∈(0,1), and consequently its quasi-Assouad dimension is equal to $1.$ These results settle an open problem on dimension of graphs posed by Fraser.
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BibTeX
@misc{shah2026assouad,
title = {{Assouad type dimensions of generalized affine fractal interpolation functions and their applications}},
author = {Aaryan Dharmesh Shah and Sangita Jha},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.39524},
eprint = {2609.39524},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.39524},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Assouad type dimensions of generalized affine fractal interpolation functions and their applications
AU - Aaryan Dharmesh Shah
AU - Sangita Jha
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.39524
UR - https://arxiv.org/abs/2609.39524
AB - In this article, we investigate the Assouad spectrum and Assouad dimension of graphs of fractal functions generated by generalized affine iterated function systems. We establish upper and lower bounds for the Assouad spectrum in terms of the scaling functions and the underlying partition of the generalized affine construction. If the scaling function is Lipschitz continuous and the partition is uniform, we obtain an explicit expression for the Assouad spectrum of the associated graph. These results provide a connection between the parameters defining the generalized affine fractal function and the local multiscale geometry of its graph. As applications, we consider two classic examples of generalized affine fractal functions, namely the Weierstrass and Takagi functions. For the classical Weierstrass function W, whose graph Γ_W has the box dimension 2+log_Nλ, we obtain dim_A^θ(Γ_W) ≤ (2+log_Nλ−θ)/(1−θ), θ∈ (0,log_N1/λ). and if λ^2 N <1, we get dim_A(Γ_W)≥ 1 +log_N(1/λ). For the classical Takagi function T with graph Γ_T, we show that dim_A^θ(Γ_T)=1, θ∈(0,1), and consequently its quasi-Assouad dimension is equal to $1.$ These results settle an open problem on dimension of graphs posed by Fraser.
ER -