Nonexistence of Frame Measures for Separated Uniform Consecutive-Digit Bernoulli Convolutions

Abstract

We investigate the existence problem of frame measures for uniform consecutive-digit Bernoulli convolutions under the separated scaling condition. Given parameters $N\ge 2$ and $0<\rho <1/N$, we prove that if ${\rho }^{-m}=B$ for some integers $m\ge 1$ and $B\ge 2$, with $N\nmid B$, then the associated $N$-Bernoulli convolution ${\mu }_{\rho ,N}$ admits no frame measure. The proof introduces a new cyclic mask-quotient obstruction adapted to the multi-character structure of the consecutive-digit framework.

14 pages