AI 中文总结
研究对称双分支伯努利卷积的傅里叶框架存在性，引入沃尔什商障碍，通过有限坐标沃尔什包转化不等式并利用尺度关系，证明\(L^2(\mu_{\rho,d})\)在特定条件下不存在傅里叶框架，解决相关长期问题。
AI 中文摘要
我们引入一种沃尔什商障碍来研究对称双分支伯努利卷积的傅里叶框架存在性。假设\(0 < \rho < \frac{1}{2}\)且\(\rho^{-m} = B\)，其中\(m \geq 1\)为整数且\(B \geq 3\)为奇数整数。我们证明\(L^2(\mu_{\rho,d})\)不存在傅里叶框架。对于\(m = 1\)，我们的论证独立证明了奇数基康托测度的非框架定理，从而解决了斯特里赫兹关于中间三分之一康托测度的长期未解决问题。对于\(m > 1\)，我们的定理处理非整数倒数幂收缩率\(\rho = B^{-1/m}\)，这超出了经典整数基康托测度的范围。我们的证明是自包含的。它使用有限坐标沃尔什包将框架不等式转化为不相容的切线商估计，而\(\rho^{-m} = B\)提供了精确的\(m\)步尺度关系导致矛盾。
英文摘要
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 $0<ρ<\frac12$ and $ρ^{-m}=B$ for some integer $m\ge1$ and odd integer $B\ge3$. We prove that $L^2(μ_{ρ,d})$ 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 $ρ=B^{-1/m}$, 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 $ρ^{-m}=B$ supplies the exact $m$-step scale relation leading to the contradiction.
Comments15 pages. Revised exposition and proof presentation; corrected notation, references, and minor errors. Main results unchanged
