Local Extrema and Infinite Derivatives of the Delange Function of Base 3

Source abstract

In the ternary Hamming cube $Q_n^3$, the smallest number $ξ_n(m)$ of edges leaving a set of $m$ vertices is $2(nm-S_3(m))$, where $S_3(m)=\sum_{j<m}s_3(j)$ and $s_3$ is the ternary digit sum. After rescaling, these numbers are the values at the points $m/3^n$ of a continuous function $T$ of Takagi type, the Delange function of base $3$. Both facts are known. In this paper, we study the local structure of $T$. The local minima of $T$ are exactly the ternary rationals, that is, the numbers $m/3^n$ with integers $m$ and $n\ge0$. The local maxima are exactly the points $(m+1/2)/3^n+k$, where $k$, $m$ and $n$ are integers with $n\ge0$, $0\le m<3^n$ and $s_3(m)=n$. We also characterize the points $x$ where $T$ has an infinite derivative from one or both sides, in terms of their ternary digits $ξ_1,ξ_2,\ldots$. At a ternary rational the derivative is $+\infty$ from the right and $-\infty$ from the left. At every other point, the derivative from the right is $+\infty$ exactly when the digit slopes $2\sum_{j\le n}(1-ξ_j)$ tend to $+\infty$. The derivative from the left is $+\infty$ exactly when a stronger condition holds, which also involves the lengths of the maximal strings of consecutive zeros among the digits $ξ_j$. For the value $-\infty$ the two sides are exchanged, and twos replace zeros. 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. For the finite cubes, we show that over every block of sizes $a3^k\le m\le(a+1)3^k$, the maximum of $ξ_n(m)$ exceeds the larger of its values at the two ends of the block by exactly $\max\{3^{k-|d|}-1,0\}$, where $d=n-k-s_3(a)$. Finally, the variation $V_n$ of $T$ over the partition of $[0,1]$ into the $3^n$ intervals $[m/3^n,(m+1)/3^n]$ satisfies $V_n/\sqrt n\to4/\sqrt{3π}$, and we compute the Fourier coefficients of $T$.