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

Abstract

A finite Borel measure $μ$ in ${\mathbb R}^d$ is called a frame-spectral measure if it admits an exponential frame (or Fourier frame) for $L^2(μ)$. 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 $ν\ast λ+δ_t\astν$ 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 $μ_4+μ_{16}$ 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 $R_θ$ with $θ\ne \pmπ/2$, the two-dimensional measure $ρ_θ (\cdot): = (μ_4\timesδ_0)(\cdot)+(δ_0\timesμ_{16})(R_θ^{-1}\cdot)$, supported on the union of $x$-axis and $y=(\cot θ)x$, always admit a Fourier frame. Furthermore, we can find $\{e^{2πi \langleλ,x\rangle}\}_{λ\inΛ_θ}$ such that it forms a Fourier frame for $ρ_θ$ with frame bounds independent of $θ$. Nonetheless, $ρ_{\pmπ/2}$ does not admit any Fourier frame.