On Fourier frame of absolutely continuous measures

Let μ be a compactly supported absolutely continuous probability measure on Rn, we show that L2(K,dμ) admits a Fourier frame if and only if its Radon–Nikodym derivative is bounded above and below almost everywhere on the support K. As a consequence, we prove that if μ is an equal weight absolutely continuous self-similar measure on R1 and L2(K,dμ) admits a Fourier frame, then the density of μ must be a characteristic function of self-similar tile. In particular, this shows for almost everywhere 1/2<λ<1, the L2 space of the λ-Bernoulli convolutions cannot admit a Fourier frame.