Sweetened and flavored dessert made from gelatinous or starchy ingredients and milk

Takagi (高木) curves are fractals that are somehow less known than Cantor sets and Sierpinski carpets, yet they can also be useful as (counter-)examples. The general 高木 curve is the graph $y=f(x)$ of a function $f$ that is built from triangular waves. The $n$th generation wave has equation $y=2^{-n} \lbrace 2^n x \rbrace$ where $\lbrace\cdot\rbrace$ means the distance to the nearest integer. Six of these waves are pictured below.

Summation over $n$ creates the standard 高木 curve $T$, also known as the blancmange curve:

$\displaystyle y=\sum_{n=0}^{\infty} 2^{-n} \lbrace 2^n x\rbrace$

Note the prominent cusps at dyadic rationals: more on this later.

General 高木 curves are obtained by attaching coefficients $c_n$ to the terms of the above series. The simplest of these, and the one of most interest to me, is the alternating 高木 curve $T_{alt}$:

$\displaystyle y=\sum_{n=0}^{\infty} (-2)^{-n} \lbrace 2^n x\rbrace$

The alternation of signs destroys the cusps that are so prominent in $T$. Quantitatively speaking, the diameter of any subarc of $T_{alt}$ is bounded by the distance between its endpoints times a fixed constant. The curves with this property are called quasiarcs, and they are precisely the quasiconformal images of line segments.

Both $T$ and $T_{alt}$ have infinite length. More precisely, the length of the $n$th generation of either curve is between $\sqrt{(n+1)/2}$ and $\sqrt{n+1}+1$. Indeed, the derivative of $x\mapsto 2^{-k}\lbrace 2^k x\rbrace$ is just the Rademacher function $r_k$. Therefore, the total variation of the sum $\sum_{k=0}^n c_k 2^{-k}\lbrace 2^k x\rbrace$ is the $L^1$ norm of $\sum_{k=0}^n c_k r_k$. With $c_k=\pm 1$ the sharp form of the Хинчин inequality from the previous post yields

$\displaystyle 2^{-1/2}\sqrt{n+1} \le \left\|\sum_{k=0}^n c_k r_k\right\|_{L^1} \le \sqrt{n+1}$

For the upper bound I added 1 to account for the horizontal direction. Of course, the bound of real interest is the lower one, which proves unrectifiability. So far, a construction involving these curves shed a tiny bit of light on the following questions:

Which sets $K\subset \mathbb R^n$ have the property that any quasiconformal image of $K$ contains a rectifiable curve?

I won’t go (yet) into the reasons why this question arose. Any set with nonempty interior has the above property, since quasiconformal maps are homeomorphisms. A countable union of lines in the plane does not; this is what 高木 curves helped to show. The wide gap between these results remains to be filled.