Translational absolute continuity and Fourier frames on a sum of singular measures

A finite Borel measure μ in Rd is called a frame-spectral measure if it admits an exponential frame (or Fourier frame) for L2(μ). It has been conjectured that a frame-spectral measure must be translationally absolutely continuous, which is a criterion describing the local uniformity of a measure on its support. In this paper, we show that if any measures ν and λ without atoms whose supports form a packing pair, then ν⁎λ+δt⁎ν is translationally singular and it does not admit any Fourier frame. In particular, we show that the sum of one-fourth and one-sixteenth Cantor measure μ4+μ16 does not admit any Fourier frame. We also interpolate the mixed-type frame-spectral measures studied by Lev and the measure we studied. In doing so, we demonstrate a discontinuity behavior: For any anticlockwise rotation mapping Rθ with θ≠±π/2, the two-dimensional measure ρθ(⋅):=(μ4×δ0)(⋅)+(δ0×μ16)(Rθ−1⋅), supported on the union of x-axis and y=(cot⁡θ)x, always admit a Fourier frame. Furthermore, we can find {e2πi〈λ,x〉}λ∈Λθ such that it forms a Fourier frame for ρθ with frame bounds independent of θ. Nonetheless, ρ±π/2 does not admit any Fourier frame.