Local Extrema and Infinite Derivatives of the Delange Function of Base 3
Abstract
The Delange function of base $3$ is the continuous function T(x)=∑j≥03−jτ(3jx) of Takagi type, where τ(x)=2min{dist(x,ℤ),1/3}. Multiplied by 3n, its values at the points m/3n are the smallest numbers ξn(m)=2(nm−S3(m)) of edges leaving a set of m vertices in the ternary Hamming cube Qn3, where S3(m)=∑j<ms3(j) and s3 is the ternary digit sum. Both facts are known. We show that the local minima of T are exactly the ternary rationals m/3n, and that the local maxima are exactly the points (m+1/2)/3n+k with integers k, m, n such that n≥0, 0≤ m<3n and s3(m)=n. We also characterize, in terms of the ternary digits ξ1,ξ2,… of x, when T has an infinite derivative at x from one or both sides. At a ternary rational the right derivative is +∞ and the left derivative is −∞. At every other point, the right derivative is +∞ exactly when ∑j≤ n(1−ξj)→+∞. The condition for the left derivative to be +∞ is stronger and also involves the lengths of the maximal strings of zeros among the digits. For the value −∞ the two sides are exchanged and the roles of zeros and twos are swapped. In particular, T has an infinite derivative at every rational point that is not a ternary rational and whose period has different numbers of zeros and twos. We also determine the maximum of ξn(m) over every block of sizes a3k≤ m≤(a+1)3k, describe by the ternary digits of h the sizes h for which the h-extra edge connectivity of Qn3 equals ξn(h), and compute the Fourier coefficients of T and its variation at each level. Finally, we determine the level set at height 3/4, which has dimension log2/log9, and show that no level set has a larger upper box dimension and that {x∈[0,1]:T(x)=y} is finite for almost every y.
Related
BibTeX
@misc{zhu2026local,
title = {{Local Extrema and Infinite Derivatives of the Delange Function of Base 3}},
author = {Chengyi Zhu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.21265},
eprint = {2610.21265},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.21265},
}
RIS
TY - UNPB
TI - Local Extrema and Infinite Derivatives of the Delange Function of Base 3
AU - Chengyi Zhu
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.21265
UR - https://arxiv.org/abs/2610.21265
AB - The Delange function of base $3$ is the continuous function T(x)=∑_j≥03^−jτ(3^jx) of Takagi type, where τ(x)=2min{dist(x,ℤ),1/3}. Multiplied by 3^n, its values at the points m/3^n are the smallest numbers ξ_n(m)=2(nm−S_3(m)) of edges leaving a set of m vertices in the ternary Hamming cube Q_n^3, where S_3(m)=∑_j<ms_3(j) and s_3 is the ternary digit sum. Both facts are known. We show that the local minima of T are exactly the ternary rationals m/3^n, and that the local maxima are exactly the points (m+1/2)/3^n+k with integers k, m, n such that n≥0, 0≤ m<3^n and s_3(m)=n. We also characterize, in terms of the ternary digits ξ_1,ξ_2,… of x, when T has an infinite derivative at x from one or both sides. At a ternary rational the right derivative is +∞ and the left derivative is −∞. At every other point, the right derivative is +∞ exactly when ∑_j≤ n(1−ξ_j)→+∞. The condition for the left derivative to be +∞ is stronger and also involves the lengths of the maximal strings of zeros among the digits. For the value −∞ the two sides are exchanged and the roles of zeros and twos are swapped. In particular, T has an infinite derivative at every rational point that is not a ternary rational and whose period has different numbers of zeros and twos. We also determine the maximum of ξ_n(m) over every block of sizes a3^k≤ m≤(a+1)3^k, describe by the ternary digits of h the sizes h for which the h-extra edge connectivity of Q_n^3 equals ξ_n(h), and compute the Fourier coefficients of T and its variation at each level. Finally, we determine the level set at height 3/4, which has dimension log2/log9, and show that no level set has a larger upper box dimension and that {x∈[0,1]:T(x)=y} is finite for almost every y.
ER -