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 $ρ^{-m}=B$ 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.

Disclosure

“he lower frame inequality for the sequence {Fs }. Therefore, no frame measure Γ for µ can exist. Declaration of Generative AI and AI-Assisted Tech- nologies in the Manuscript Preparation Process During the preparation of this manuscript, generative AI was used for linguistic polishing of the introduction and for checking the formatting of the reference list. All authors thoroughly reviewed, revised, and finalized the manuscript and take full responsibility for the content of the article”

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2026-08-22T18:14:18.369868Z

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