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Local level sets of the Takagi-van der Waerden function
Abstract
In this paper, we investigate the Takagi-van der Waerden function, Tr(x) = ∑n=0∞ (ϕ(rn x))/rn , x∈ [0,1], r ≥ 2, where ϕ(x)=dist(x,ℤ) represents the distance from x to the nearest integer. Lagarias and Maddock[13] introduced the notion of local level sets for the classical Takagi function T2. They proved that if y is uniformly distributed over the range of T2, then the expected number of local level sets contained in the level set at height y equals 3/2. We define the corresponding local level sets for every integer r≥2 and prove that, if y is uniformly distributed over the range of Tr, then the expected number of local level sets contained in the level set at height y equals (r+1)/r if r is even, and (r+1)/(r−1) if r is odd.
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BibTeX
@misc{jiang2026local,
title = {{Local level sets of the Takagi-van der Waerden function}},
author = {Lai Jiang and Ting-Ting Ying and Yi-Yang Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.08572},
eprint = {2610.08572},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.08572},
}
RIS
TY - UNPB
TI - Local level sets of the Takagi-van der Waerden function
AU - Lai Jiang
AU - Ting-Ting Ying
AU - Yi-Yang Zhang
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.08572
UR - https://arxiv.org/abs/2610.08572
AB - In this paper, we investigate the Takagi-van der Waerden function, T_r(x) = ∑_n=0^∞ (ϕ(r^n x))/r^n , x∈ [0,1], r ≥ 2, where ϕ(x)=dist(x,ℤ) represents the distance from x to the nearest integer. Lagarias and Maddock[13] introduced the notion of local level sets for the classical Takagi function T_2. They proved that if y is uniformly distributed over the range of T_2, then the expected number of local level sets contained in the level set at height y equals 3/2. We define the corresponding local level sets for every integer r≥2 and prove that, if y is uniformly distributed over the range of T_r, then the expected number of local level sets contained in the level set at height y equals (r+1)/r if r is even, and (r+1)/(r−1) if r is odd.
ER -