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 nth 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 nth 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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