arXiv subjects

Xiao-Ye Fu

Publications and source records attributed to Xiao-Ye Fu.

4 recordsLinked to original sources

arXiv2026

Nonexistence of Frame Measures for Separated Uniform Consecutive-Digit Bernoulli Convolutions

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

math.FA↗

arXiv2026

Tur\'an problem on the union of three intervals of equal length

This paper addresses the Tur\'an extremal problem on the symmetric three-interval set $\Omega_\lambda = (-1,1) \cup (\lambda-1,\lambda+1) \cup (-\lambda-1,-\lambda+1)$, $\lambda \geq 0$. We establish a discrete approximation principle relating the continuous Tur\'an problem on $\Omega_\lambda$ to the discrete Tur\'an problem on an associated finite discrete set. This enables the continuous problem to be exactly expressed as a limit of finite nonnegative trigonometric polynomial problems, which are equivalent to finite semidefinite programs. Using this framework, we compute the Tur\'an constant for integer and rational parameters $\lambda$. We further analyze the dependence of the Tur\'an constant on $\lambda$ and derive upper bounds for all $\lambda > 5$.

math.CA↗

arXiv2026

Rational Spectra for Finite Hadamard Pairs and Bounded Spectral Sets on the real line

We prove that every finite spectral pair $(A,\Gamma)$ with $A\subset\mathbb Z$, $\Gamma\subset\mathbb R/\mathbb Z$, and $0\in\Gamma$ has a rational spectrum, that is, $\Gamma\subset\mathbb Q/\mathbb Z$. Our argument relies on a modulus rigidity theorem for generalizedVandermonde systems satisfying inverse-orthogonality relations, constructed by hyperbolic positive-definite kernels. Combined with Galois conjugation and Kronecker's theorem, this rigidity forces the associated exponential nodes to be roots of unity. As an application, using the periodicity and fiberization of one-dimensional spectra, we show that every spectrum $\Lambda$ of a bounded measurable spectral set $\Omega\subset\mathbb R$ with $|\Omega|=1$ and $0\in\Lambda$ is contained in $\mathbb Q$. This rational-spectrum result completes a chain of equivalences proven by Dutkay and Lai, which reduces the full one-dimensional Fuglede's conjecture to its finite cyclic analogues over $\mathbb{Z}_n$.

math.FA↗

arXiv2026

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

We introduce a Walsh--quotient obstruction to study Fourier-frame existence for symmetric two-branch Bernoulli convolutions \[ \mu_{\rho,d} =\ast_{j=1}^{\infty} \frac12\bigl(\delta_{-d\rho^{j}/2}+\delta_{d\rho^{j}/2}\bigr), \qquad 0<\rho<1,\quad d>0. \] Suppose that $0<\rho<\frac12$ and $\rho^{-m}=B$ for some integer $m\ge1$ and odd integer $B\ge3$. 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.

math.FA↗