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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.

Triangular Waves

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

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

Standard Takagi curve

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 $T_{alt}$ :

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

Alternating Takagi curve

This GeoGebra applet shows several generations of $T_{alt}$ . 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.

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