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 $0&lt;ρ&lt;\frac12$ and $ρ<sup>{-m}=B$ for some integer $m\ge1$ and odd integer $B\ge3$. We prove that $L<sup>2(μ_{ρ,d})$ 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 $m&gt;1$, our theorem includes the non-integer reciprocal-power contraction ratios $ρ=B<sup>{-1/m}$, 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 $ρ<sup>{-m}=B$ supplies the exact $m$-step scale relation leading to the contradiction.

• The paper establishes that L² spaces for Bernoulli convolutions with contraction parameters ρ satisfying ρ⁻ᵐ = B (with B odd) do not support unweighted exponential Fourier frames.
• It employs the novel Walsh-quotient obstruction, translating frame inequalities into recurrence relations of tangent quotients using Walsh packet functionals.
• The result resolves Strichartz's question for the middle-third Cantor measure and extends the finding to non-integer contraction cases.

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

Introduction and Problem Setting

This work presents an obstruction to the existence of Fourier frames for $L^2$ spaces over symmetric two-branch Bernoulli convolutions with contraction parameters of a specific algebraic form. The analysis is motivated by Strichartz's well-known question regarding the existence of Fourier frames for the middle-third Cantor measure and, more generally, self-similar singular measures where spectrality is absent but stable non-orthogonal expansions may still exist.

The measures in question are

$$\mu_{\rho,d} = \ast_{j=1}^{\infty} \tfrac12\left( \delta_{-d\rho^j/2} + \delta_{d\rho^j/2} \right), \qquad 0 < \rho < 1, \ d > 0,$$

or, equivalently, random series distributions

$\sum_{j=1}^\infty \tfrac{d\xi_j\rho^j}{2}$

with $\xi_j \in \{-1,1\}$ i.i.d. symmetric. The classical setting includes the standard Cantor measures as special cases. The main result establishes that if the contraction parameter $\rho$ satisfies $\rho^{-m} = B$ for some integer $m \geq 1$ and odd integer $B\geq 3$, then $L^2(\mu_{\rho,d})$ does not admit any exponential Fourier frame.

For $m = 1$, this resolves Strichartz's question, including for the archetype middle-third Cantor measure, by proving the frame property fails for all odd-integer-base Cantor measures—a problem open for over two decades. Importantly, the result extends beyond the integer case ($$\mu_{\rho,d} = \ast_{j=1}^{\infty} \tfrac12\left( \delta_{-d\rho^j/2} + \delta_{d\rho^j/2} \right), \qquad 0 < \rho < 1, \ d > 0,$$0) to genuinely non-integer contractions ($$\mu_{\rho,d} = \ast_{j=1}^{\infty} \tfrac12\left( \delta_{-d\rho^j/2} + \delta_{d\rho^j/2} \right), \qquad 0 < \rho < 1, \ d > 0,$$1 and $$\mu_{\rho,d} = \ast_{j=1}^{\infty} \tfrac12\left( \delta_{-d\rho^j/2} + \delta_{d\rho^j/2} \right), \qquad 0 < \rho < 1, \ d > 0,$$2 not an $$\mu_{\rho,d} = \ast_{j=1}^{\infty} \tfrac12\left( \delta_{-d\rho^j/2} + \delta_{d\rho^j/2} \right), \qquad 0 < \rho < 1, \ d > 0,$$3th power).

Main Technical Contribution

The paper introduces and leverages what the authors call the "Walsh-quotient obstruction," an analytic approach using Walsh packet functionals encoded via the symbolic model of the underlying random series. A key feature of the analysis is the translation of frame inequalities for exponentials into recurrence/dynamical properties of sequences of tangent quotients, exploiting the hierarchical, finite-coordinate structure on the symbolic space and careful product computations.

Given the explicit relation $$\mu_{\rho,d} = \ast_{j=1}^{\infty} \tfrac12\left( \delta_{-d\rho^j/2} + \delta_{d\rho^j/2} \right), \qquad 0 < \rho < 1, \ d > 0,$$4 with $$\mu_{\rho,d} = \ast_{j=1}^{\infty} \tfrac12\left( \delta_{-d\rho^j/2} + \delta_{d\rho^j/2} \right), \qquad 0 < \rho < 1, \ d > 0,$$5 odd, a shift in scale by $$\mu_{\rho,d} = \ast_{j=1}^{\infty} \tfrac12\left( \delta_{-d\rho^j/2} + \delta_{d\rho^j/2} \right), \qquad 0 < \rho < 1, \ d > 0,$$6 induces an exact functional equation: for the sequence $$\mu_{\rho,d} = \ast_{j=1}^{\infty} \tfrac12\left( \delta_{-d\rho^j/2} + \delta_{d\rho^j/2} \right), \qquad 0 < \rho < 1, \ d > 0,$$7,

$$\mu_{\rho,d} = \ast_{j=1}^{\infty} \tfrac12\left( \delta_{-d\rho^j/2} + \delta_{d\rho^j/2} \right), \qquad 0 < \rho < 1, \ d > 0,$$8

This "level-shift" structure, analogous to transfer operators on trigonometric multipliers, enables a rigorous propagation argument: for large $$\mu_{\rho,d} = \ast_{j=1}^{\infty} \tfrac12\left( \delta_{-d\rho^j/2} + \delta_{d\rho^j/2} \right), \qquad 0 < \rho < 1, \ d > 0,$$9, $\sum_{j=1}^\infty \tfrac{d\xi_j\rho^j}{2}$0 is also necessarily large, and their product grows uniformly—leading to a contradiction with frame bounds when tested against appropriately constructed Walsh packets.

The proof establishes that:

• Testing the frame inequalities on one- and two-coordinate Walsh packets allows the expression of frame-induced $\sum_{j=1}^\infty \tfrac{d\xi_j\rho^j}{2}$1-norms in terms of these tangent functions.
• For $\sum_{j=1}^\infty \tfrac{d\xi_j\rho^j}{2}$2 with $\sum_{j=1}^\infty \tfrac{d\xi_j\rho^j}{2}$3 odd, the tangent recursion forces either decay or blow-up, incompatible with the existence of frame bounds.
• The non-overlap condition $\sum_{j=1}^\infty \tfrac{d\xi_j\rho^j}{2}$4 is necessary for the injectivity of the symbolic code and the direct transfer of combinatorial Walsh packets to $\sum_{j=1}^\infty \tfrac{d\xi_j\rho^j}{2}$5.

Strong and Contradictory Claims

The principal claim is that $\sum_{j=1}^\infty \tfrac{d\xi_j\rho^j}{2}$6 admits no (unweighted) exponential Fourier frame whenever $\sum_{j=1}^\infty \tfrac{d\xi_j\rho^j}{2}$7 for any odd integer $\sum_{j=1}^\infty \tfrac{d\xi_j\rho^j}{2}$8, $\sum_{j=1}^\infty \tfrac{d\xi_j\rho^j}{2}$9, and $\xi_j \in \{-1,1\}$0. In particular, this covers all odd-integer-base and a swath of non-integer reciprocal-power contractions otherwise beyond the scope of previous results. For $\xi_j \in \{-1,1\}$1, this categorically settles the open problem for the middle-third Cantor measure and all its odd-integer extensions, asserting the complete nonexistence of such discrete Fourier frames in $\xi_j \in \{-1,1\}$2 for these measures.

This is in marked contrast to the even-integer case, where Dai's theorem provides existence (and full spectrality). Thus, there is a sharp dichotomy at the parity of the base, which the analytic approach here renders transparent in terms of tangent-dynamical obstruction.

Context and Implications

Previous results addressed spectrality for Cantor-type and Bernoulli convolution measures, but nonspectrality does not a priori preclude Fourier frames—Lai and Wang [LaiWang2017] notably constructed singular measures with only finite orthogonality, yet admitting frames. This work fills the prior gap for frame existence, affirming that, at least for odd reciprocal bases, not only does orthogonality fail, but stable expansions cannot be achieved even in the significantly weaker sense of Fourier frames.

Practically, this result closes the possibility of constructing such frames (and any derived harmonic analysis machinery) for a broad class of singular measures central to applications in analysis on fractals, signal processing on Cantor sets, and non-linear self-similar measures. Theoretically, it highlights the subtle interactions between arithmetic structure, recursive self-similarity, and harmonic analytic properties, with the parity-odd scale step leading to inescapable analytic obstruction.

Extensions to the overlapping regime ($\xi_j \in \{-1,1\}$3) or contraction parameters outside the precise algebraic relation considered would require new ideas, as the symbolic-coding injectivity and dynamical recursion central to the proof collapse in general. The authors conjecture a full equivalence between spectrality and frame-existence for Bernoulli convolutions in the non-overlapping regime, a natural direction for future inquiry.

Outlook for Future Research

The analytic and combinatorial apparatus developed here suggests a promising framework for addressing Fourier-analytic properties of other self-affine, random, or measure-theoretically exotic objects. Future developments may include:

• Extending obstruction criteria to overlapping or more general self-similar measures via ergodic, dynamical, or random matrix techniques.
• Investigating the precise properties of tangent quotient dynamics in non-algebraic or Diophantine contraction regimes.
• Studying weighted or generalized frames, as in Parseval weighted exponentials, where this obstruction may not apply.

The delineation of the Fourier frame property for singular measures thus remains an active, highly technical area, with this work providing a decisive advance on a foundational open question.

Conclusion

This paper establishes a rigorous obstruction to the existence of unweighted exponential Fourier frames for $\xi_j \in \{-1,1\}$4 spaces associated with odd reciprocal-power, non-overlapping Bernoulli convolutions. The result resolves Strichartz's long-standing open question for the middle-third Cantor measure and extends to a broader class of non-integer contraction ratios, synthesizing symbolic, dynamical, and analytic techniques via the Walsh-quotient method. The findings enforce a sharp arithmetic dichotomy for Fourier frames on singular measures and point toward the necessity of new methods for any further generalization.