arXiv subjects

Xiaoye Fu

Publications and source records attributed to Xiaoye Fu.

6 recordsLinked to original sources

arXiv2019

Open set condition and pseudo Hausdorff measure of self-affine IFSs

Let $A$ be an $n\times n$ real expanding matrix and $\mathcal{D}$ be a finite subset of $\mathbb{R}^n$ with $0\in\mathcal{D}$. The family of maps $\{f_d(x)=A^{-1}(x+d)\}_{d\in\mathcal{D}}$ is called a self-affine iterated function system (self-affine IFS). The self-affine set $K=K(A,\mathcal{D})$ is the unique compact set determined by $(A, {\mathcal D})$ satisfying the set-valued equation $K=\displaystyle\bigcup_{d\in\mathcal{D}}f_d(K)$. The number $s=n\,\ln(\# \mathcal{D})/\ln(q)$ with $q=|\det(A)|$, is the so-called pseudo similarity dimension of $K$. As shown by He and Lau, one can associate with $A$ and any number $s\ge 0$ a natural pseudo Hausdorff measure denoted by $\mathcal{H}_w^s.$ In this paper, we show that, if $s$ is chosen to be the pseudo similarity dimension of $K$, then the condition $\mathcal{H}_w^s(K)> 0$ holds if and only if the IFS $\{f_d\}_{d\in\mathcal{D}}$ satisfies the open set condition (OSC). This extends the well-known result for the self-similar case that the OSC is equivalent to $K$ having positive Hausdorff measure $\mathcal{H}^s$ for a suitable $s$. Furthermore, we relate the exact value of pseudo Hausdorff measure $\mathcal{H}_w^s(K)$ to a notion of upper $s$-density with respect to the pseudo norm $w(x)$ associated with $A$ for the measure $μ=\lim\limits_{M\to\infty}\sum\limits_{d_0,\dotsc,d_{M-1}\in\mathcal{D}}δ_{d_0 + Ad_1 + \dotsb + A^{M-1}d_{M-1}}$ in the case that $\#\mathcal{D}\le\lvert\det A\rvert$.

math.MG↗

arXiv2019

On Spectral Cantor-Moran measures and a variant of Bourgain's sum of sine problem

In this paper, we show that if we have a sequence of Hadamard triples $\{(N_n,B_n,L_n)\}$ with $B_n\subset \{0,1,..,N_n-1\}$ for $n=1,2,...$, except an extreme case, then the associated Cantor-Moran measure $$ \begin{aligned} μ= μ(N_n,B_n) =& δ_{\frac{1}{N_1}B_1}\astδ_{\frac{1}{N_1N_2}B_2}\ast δ_{\frac{1}{N_1N_2N_3}B_3}\ast...\\ =& μ_n\astμ_{>n} \end{aligned} $$ with support inside $[0,1]$ always admits an exponential orthonormal basis $E(Λ) = \{e^{2πi λx}:λ\inΛ\}$ for $L^2(μ)$, where $Λ$ is obtained from suitably modifying $L_n$. Here, $μ_n$ is the convolution of the first $n$ Dirac measures and $μ_{>n}$ denotes the tail-term. We show that the completeness of $E(Λ)$ in general depends on the ``equi-positivity" of the sequence of the pull-backed tail of the Cantor-Moran measure $ν_{>n}(\cdot) = μ_{>n}((N_1...N_n)^{-1}(\cdot))$. Such equi-positivity can be analyzed by the integral periodic zero set of the weak limit of $\{ν_{>n}\}$. This result offers a new conceptual understanding of the completeness of exponential functions and it improves significantly many partial results studied by recent research, whose focus has been specifically on $\#B_n\le 4$. Using the Bourgain's example that a sum of sine can be asymptotically small, we shows that, in the extreme case, there exists some Cantor-Moran measure such that the equi-positive condition fails and the Fourier transform of the associated $ν_{>n}$ uniformly converges on some unbounded set.

math.CA↗

arXiv2017

Translational absolute continuity and Fourier frames on a sum of singular measures

A finite Borel measure $μ$ in ${\mathbb R}^d$ is called a frame-spectral measure if it admits an exponential frame (or Fourier frame) for $L^2(μ)$. It has been conjectured that a frame-spectral measure must be translationally absolutely continuous, which is a criterion describing the local uniformity of a measure on its support. In this paper, we show that if any measures $ν$ and $λ$ without atoms whose supports form a packing pair, then $ν\ast λ+δ_t\astν$ is translationally singular and it does not admit any Fourier frame. In particular, we show that the sum of one-fourth and one-sixteenth Cantor measure $μ_4+μ_{16}$ does not admit any Fourier frame. We also interpolate the mixed-type frame-spectral measures studied by Lev and the measure we studied. In doing so, we demonstrate a discontinuity behavior: For any anticlockwise rotation mapping $R_θ$ with $θ\ne \pmπ/2$, the two-dimensional measure $ρ_θ (\cdot): = (μ_4\timesδ_0)(\cdot)+(δ_0\timesμ_{16})(R_θ^{-1}\cdot)$, supported on the union of $x$-axis and $y=(\cot θ)x$, always admit a Fourier frame. Furthermore, we can find $\{e^{2πi \langleλ,x\rangle}\}_{λ\inΛ_θ}$ such that it forms a Fourier frame for $ρ_θ$ with frame bounds independent of $θ$. Nonetheless, $ρ_{\pmπ/2}$ does not admit any Fourier frame.

math.FA↗

arXiv2016

Decomposition of Integral Self-Affine Multi-Tiles

In this paper, we propose a method to decompose an integral self-affine ${\mathbb Z}^n$-tiling set $K$ into measure disjoint pieces $K_j$ satisfying $K=\displaystyle\bigcup K_j$ in such a way that the collection of sets $K_j$ forms an integral self-affine collection associated with the matrix $B$ and this with a minimum number of pieces $K_j$. When used on a given measurable $\mathbb{Z}^n$-tiling set $K\subset\mathbb{R}^n$, this decomposition terminates after finitely many steps if and only if the set $K$ is an integral self-affine multi-tile. Furthermore, we show that the minimal decomposition we provide is unique.

math.FA↗

arXiv2013

Spectrality of Self-Similar Tiles

We call a set $K \subset {\mathbb R}^s$ with positive Lebesgue measure a {\it spectral set} if $L^2(K)$ admits an exponential orthonormal basis. It was conjectured that $K$ is a spectral set if and only if $K$ is a tile (Fuglede's conjecture). Despite the conjecture was proved to be false on ${\mathbb R}^s$, $s\geq 3$ ([T], [KM2]), it still poses challenging questions with additional assumptions. In this paper, our additional assumption is self-similarity. We study the spectral properties for the class of self-similar tiles $K$ in ${\mathbb R}$ that has a product structure on the associated digit sets. We show that any strict product-form tiles and the associated modulo product-form tiles are spectral sets. As for the converse question, we give a pilot study for the self-similar set $K$ generated by arbitrary digit sets with four elements. We investigate the zeros of its Fourier transform due to the orthogonality, and verify Fuglede's conjecture for this special case.

math.FA↗

arXiv2013

Measure of Self-Affine Sets and Associated Densities

Let $B$ be an $n\times n$ real expanding matrix and $\mathcal{D}$ be a finite subset of $\mathbb{R}^n$ with $0\in\mathcal{D}$. The self-affine set $K=K(B,\mathcal{D})$ is the unique compact set satisfying the set-valued equation $BK=\displaystyle\bigcup_{d\in\mathcal{D}}(K+d)$. In the case where $\text{card}(\mathcal{D})=\lvert\det B\rvert,$ we relate the Lebesgue measure of $K(B,\mathcal{D})$ to the upper Beurling density of the associated measure $μ=\lim\limits_{s\to\infty}\sum\limits_{\ell_0,\dotsc,\ell_{s-1}\in\mathcal{D}}δ_{\ell_0+B\ell_1+\dotsb+B^{s-1}\ell_{s-1}}.$ If, on the other hand, $\text{card}(\mathcal{D})<\lvert\det B\rvert$ and $B$ is a similarity matrix, we relate the Hausdorff measure $\mathcal{H}^s(K)$, where $s$ is the similarity dimension of $K$, to a corresponding notion of upper density for the measure $μ$.

math.FA↗