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 $N\geq 2$ and $0<ρ<1/N$, we prove that if $ρ<sup>{-m}=B$ for some integers $m\geq 1$ and $B\geq 2$, with $N\nmid B$, then the associated $N$-Bernoulli convolution $μ_{ρ,N}$ admits no frame measure. The proof introduces a new cyclic mask-quotient obstruction adapted to the multi-character structure of the consecutive-digit framework.