Almost-everywhere finiteness of level sets

Determine whether the level set L(y)={x∈[0,1]:T(x)=y} of the base-3 Delange function is finite for almost every y∈[0,1].

Background

The authors discuss level sets of the base-3 Delange function T and exhibit a nontrivial fractal subset of L(3/4). In analogy with results for Takagi’s function, they leave unresolved whether typical level sets of T contain only finitely many points.

References

We do not know whether every level set of T has Hausdorff dimension at most $\log2/\log9$, or whether $L(y)$ is finite for almost every $y$.

— Local Extrema and Infinite Derivatives of the Delange Function of Base 3  (2610.21265 - Zhu, 15 Oct 2026) in Section “Other bases and further questions”