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Takagi-van der Waerden Function

The Takagi-van der Waerden function is defined by

T_r(x)=sum_(n=0)^infty(phi(r^nx))/(r^n),

where r>=2 is an integer, x in [0,1], and phi(u) is the distance from u to the nearest integer. For r=2, this is the blancmange function.

Jiang et al. (2026) extended the digit-based definition of a local level set to every r>=2. If the height y is chosen uniformly from the range of T_r, the expected number of local level sets contained in {x:T_r(x)=y} is (r+1)/r when r is even and (r+1)/(r-1) when r is odd (Jiang et al. 2026).

See also

Blancmange Function, Local Level Set

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References

Jiang, L.; Ying, T.-T.; and Zhang, Y.-Y. "Local Level Sets of the Takagi-van der Waerden Function." 6 Oct 2026. https://arxiv.org/abs/2610.08572.

Cite this as:

Weisstein, Eric W. "Takagi-van der Waerden Function." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Takagi-vanderWaerdenFunction.html

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