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.