Let W_a,b(t)=∑_n=0^∞a^n cos(b^n t) be a classical Weierstrass function, with 0 1 in the usual nowhere-differentiable regime. Determine for which parameters, and for which integers k≥ 0, the occupation measure mu_W,T defined by mu_W,T(A)=operatornameLebt∈[0,T]:W_a,b(t)∈ A is absolutely continuous with respect to Lebesgue measure and has a density L_W,T∈ C^k(R). In particular, determine whether sufficiently large lacunarity of the Weierstrass frequencies implies the existence of such C^k local times.

Geman–Horowitz question on local time of Weierstrass functions

Let $W_{a,b}(t)=\sum_{n=0}^{\infty}a^n\cos(b^n t)$ be a classical Weierstrass function, with $0<a<1$ and $b>1$ in the usual nowhere-differentiable regime. Determine for which parameters, and for which integers $k\ge 0$ , the occupation measure $\mu_{W,T}$ defined by $\mu_{W,T}(A)=\operatorname{Leb}\{t\in[0,T]:W_{a,b}(t)\in A\}$ is absolutely continuous with respect to Lebesgue measure and has a density $L_{W,T}\in C^k(\mathbb{R})$ . In particular, determine whether sufficiently large lacunarity of the Weierstrass frequencies implies the existence of such $C^k$ local times.

Equivalent formulations 1Other wordings

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

Occupation-measure formulation
For $W_{a,b}$ and $T>0$ , determine when there exists $L_{W,T}\in C^k(\mathbb{R})$ such that $\int_0^T\varphi(W_{a,b}(t))\,dt=\int_{\mathbb{R}}\varphi(y)L_{W,T}(y)\,dy$ for every suitable test function $\varphi$ .
source: Arbitrarily Fast Quantum Dispersion in Long-Range Crystals

References

Primary source

arXiv

Additional references

Arbitrarily Fast Quantum Dispersion in Long-Range Crystals — arXiv

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Progress summary

Refreshed 1 month ago

Claimed progress

A new unrefereed preprint claims substantial progress on local times for highly lacunary Weierstrass functions, but the general question remains open.

The Geman–Horowitz question asks which Fourier-series functions possess sufficiently regular local times. Existing results cover generic parameter families, not every deterministic Weierstrass function; the latest preprint claims a result for sufficiently large lacunarity.

Known results

For prevalent $\alpha$ -Weierstrass functions with $b \ge 2$ and $0 < \alpha < 1$ , the occupation measure has an $L^2(\mathbb{R})$ density for almost every parameter choice (2025).
For $0 < \alpha < \frac{1}{2}$ , that density is bounded and continuous (2025).
A 2020 study obtained related absolute-continuity and square-integrability results for dynamical-system measures, but only suggested applications to local times.

August 27, 2026 claimed advance

Arbitrarily Fast Quantum Dispersion in Long-Range Crystals claims that Fourier-decay and van der Corput methods establish regular local times, together with arbitrarily fast polynomial quantum dispersion, in a sufficiently large-lacunarity regime. The preprint is unrefereed, so the claim remains unverified.

Current status (as of August 2026): Regular local times are claimed for a substantial sufficiently large-lacunarity regime, while the full Geman–Horowitz question for individual deterministic Weierstrass functions remains open and the new claim is unverified.

Sources

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Occupation-measure formulation

Primary source

Additional references

Progress summary