Classify singular measures admitting frames of exponential functions

Determine precisely the class of singular finite Borel measures on [0,1) that admit a frame of exponential functions in L^2(μ), i.e., characterize all singular measures μ for which there exists a countable set of frequencies Λ ⊂ ℝ such that {e^{2π i λ x}}_{λ∈Λ} forms a frame for L^2(μ).

Background

The paper surveys known results on Fourier frames for measures on [0,1). For absolutely continuous measures, Lai established a complete characterization: the Radon–Nikodym derivative must be bounded above and below on its support for the existence of Fourier frames of exponentials.

In contrast, several examples of singular measures with exponential frames are known, but a general characterization remains unresolved. The authors highlight that identifying exactly which singular measures admit frames of exponential functions is a longstanding open problem.

References

While there are other known measures μ that are singular which possess a frame of exponential functions such as in , when exactly singular measures have this property is still an open problem.

— Operator orbit frames and frame-like Fourier expansions  (2409.10706 - Berner et al., 2024) in Introduction, Duality in Fourier Series Expansions (third paragraph)

The existence of Fourier frames for non-spectral homogeneous self-similar measures has remained largely unresolved for more than two decades, with decisive progress only recently obtained for two-branch Bernoulli convolutions.

— Nonexistence of Frame Measures for Separated Uniform Consecutive-Digit Bernoulli Convolutions  (2608.19676 - Fu et al., 20 Aug 2026) in Section 1, Introduction