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Fourier transform of self-affine measures
Advances in Mathematics ( IF 1.7 ) Pub Date : 2020-11-01 , DOI: 10.1016/j.aim.2020.107349
Jialun Li , Tuomas Sahlsten

Suppose $F$ is a self-affine set on $\mathbb{R}^d$, $d\geq 2$, which is not a singleton, associated to affine contractions $f_j = A_j + b_j$, $A_j \in \mathrm{GL}(d,\mathbb{R})$, $b_j \in \mathbb{R}^d$, $j \in \mathcal{A}$, for some finite $\mathcal{A}$. We prove that if the group $\Gamma$ generated by the matrices $A_j$, $j \in \mathcal{A}$, forms a proximal and totally irreducible subgroup of $\mathrm{GL}(d,\mathbb{R})$, then any self-affine measure $\mu = \sum p_j f_j \mu$, $\sum p_j = 1$, $0 < p_j < 1$, $j \in \mathcal{A}$, on $F$ is a Rajchman measure: the Fourier transform $\widehat{\mu}(\xi) \to 0$ as $|\xi| \to \infty$. As an application this shows that self-affine sets with proximal and totally irreducible linear parts are sets of rectangular multiplicity for multiple trigonometric series. Moreover, if the Zariski closure of $\Gamma$ is connected real split Lie group in the Zariski topology, then $\widehat{\mu}(\xi)$ has a power decay at infinity. Hence $\mu$ is $L^p$ improving for all $1 < p < \infty$ and $F$ has positive Fourier dimension. In dimension $d = 2,3$ the irreducibility of $\Gamma$ and non-compactness of the image of $\Gamma$ in $\mathrm{PGL}(d,\mathbb{R})$ is enough for power decay of $\widehat{\mu}$. The proof is based on quantitative renewal theorems for random walks on the sphere $\mathbb{S}^{d-1}$.

中文翻译:

自仿射测度的傅立叶变换

假设 $F$ 是 $\mathbb{R}^d$, $d\geq 2$ 上的自仿射集,它不是单例,与仿射收缩相关 $f_j = A_j + b_j$, $A_j \in \mathrm{GL}(d,\mathbb{R})$, $b_j \in \mathbb{R}^d$, $j \in \mathcal{A}$,对于一些有限的 $\mathcal{A}$ . 我们证明,如果由矩阵 $A_j$ 生成的群 $\Gamma$,$j \in \mathcal{A}$,形成 $\mathrm{GL}(d,\mathbb{R })$,则任何自仿射测度 $\mu = \sum p_j f_j \mu$, $\sum p_j = 1$, $0 < p_j < 1$, $j \in \mathcal{A}$, on $ F$ 是 Rajchman 测度:傅立叶变换 $\widehat{\mu}(\xi) \to 0$ as $|\xi| \到\infty$。作为一个应用,这表明具有近端和完全不可约线性部分的自仿射集是多个三角级数的矩形多重集。而且,如果$\Gamma$ 的Zariski 闭包在Zariski 拓扑中连接实裂李群,则$\widehat{\mu}(\xi)$ 在无穷远处有功率衰减。因此,对于所有 $1 < p < \infty$ 和 $F$ 具有正傅立叶维数,$\mu$ 是 $L^p$ 改进。在维度 $d = 2,3$ 中 $\Gamma$ 的不可约性和 $\Gamma$ 在 $\mathrm{PGL}(d,\mathbb{R})$ 中的图像的非紧凑性足以进行功率衰减$\widehat{\mu}$。该证明基于球体 $\mathbb{S}^{d-1}$ 上随机游走的定量更新定理。3$ $\Gamma$ 的不可约性和 $\Gamma$ 在 $\mathrm{PGL}(d,\mathbb{R})$ 中的图像的非紧凑性足以满足 $\widehat{\mu 的功率衰减}$。该证明基于球体 $\mathbb{S}^{d-1}$ 上随机游走的定量更新定理。3$ $\Gamma$ 的不可约性和 $\Gamma$ 在 $\mathrm{PGL}(d,\mathbb{R})$ 中的图像的非紧凑性足以满足 $\widehat{\mu 的功率衰减}$。该证明基于球体 $\mathbb{S}^{d-1}$ 上随机游走的定量更新定理。
更新日期:2020-11-01
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