A Walsh-Quotient Obstruction for Fourier Frames on Odd Reciprocal-Power Bernoulli Convolutions

Abstract

We introduce a Walsh--quotient obstruction to study Fourier-frame existence for symmetric two-branch Bernoulli convolutions \[ μ_{ρ,d} =\ast_{j=1}^{\infty} \frac12\bigl(δ_{-dρ^{j}/2}+δ_{dρ^{j}/2}\bigr), \qquad 0<ρ<1,\quad d>0. \] Suppose that $0<\rho <\frac{1}{2}$ and ${\rho }^{-m}=B$ for some integer $m\ge 1$ and odd integer $B\ge 3$. We prove that $L^2({\mu }_{\rho ,d})$ admits no Fourier frame. For $m=1$, our argument proves the nonexistence of Fourier frames for odd-integer-base Cantor measures and hence resolves Strichartz's long-standing open problem for the middle-third Cantor measure. A contemporaneous independent proof of the case $m=1$ was obtained by Pont, Liehr and Taylor [arXiv:2607.08656v1]. For $m>1$, our theorem includes the non-integer reciprocal-power contraction ratios $\rho =B^{-1/m}$, which fall outside the classical integer-base Cantor-measure setting. Our proof is self-contained. It uses finite-coordinate Walsh packets to transform the frame inequalities into incompatible tangent-quotient estimates, while the identity ${\rho }^{-m}=B$ supplies the exact $m$-step scale relation leading to the contradiction.

15 pages. Revised exposition and proof presentation; corrected notation, references, and minor errors. Main results unchanged