Fourier transform of self-affine measures

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 $\mathbb{S}^{d-1}$.Comment: v2: 27 pages, updated references. Accepted to Advances in Mat

Trigonometric series and self-similar sets

Let $F$ be a self-similar set on $\mathbb{R}$ associated to contractions $f_j(x) = r_j x + b_j$, $j \in \mathcal{A}$, for some finite $\mathcal{A}$, such that $F$ is not a singleton. We prove that if $\log r_i / \log r_j$ is irrational for some $i \neq j$, then $F$ is a set of multiplicity, that is, trigonometric series are not in general unique in the complement of $F$. No separation conditions are assumed on $F$. We establish our result by showing that every self-similar measure $\mu$ on $F$ is a Rajchman measure: the Fourier transform $\widehat{\mu}(\xi) \to 0$ as $|\xi| \to \infty$. The rate of $\widehat{\mu}(\xi) \to 0$ is also shown to be logarithmic if $\log r_i / \log r_j$ is diophantine for some $i \neq j$. The proof is based on quantitative renewal theorems for random walks on $\mathbb{R}$.Comment: 18 pages, v2: improved the main theore

Kleinian Schottky groups, Patterson-Sullivan measures and Fourier decay

Let $\Gamma$ be a Zariski dense Kleinian Schottky subgroup of PSL2(C). Let $\Lambda(\Gamma)$ be its limit set, endowed with a Patterson-Sullivan measure $\mu$ supported on $\Lambda(\Gamma)$. We show that the Fourier transform $\widehat{\mu}(\xi)$ enjoys polynomial decay as $\vert \xi \vert$ goes to infinity. This is a PSL2(C) version of the result of Bourgain-Dyatlov [8], and uses the decay of exponential sums based on Bourgain-Gamburd sum-product estimate on C. These bounds on exponential sums require a delicate non-concentration hypothesis which is proved using some representation theory and regularity estimates for stationary measures of certain random walks on linear groups.Comment: 2 figure

Exponential mixing of geodesic flows for geometrically finite hyperbolic manifolds with cusps

Let $\Gamma$ be a geometrically finite discrete subgroup in $\operatorname{SO}(d+1,1)^{\circ}$ with parabolic elements. We establish exponential mixing of the geodesic flow on the unit tangent bundle $\operatorname{T}^1(\Gamma\backslash \mathbb{H}^{d+1})$ with respect to the Bowen-Margulis-Sullivan measure, which is the unique probability measure on $\operatorname{T}^1(\Gamma\backslash \mathbb{H}^{d+1})$ with maximal entropy. As an application, we obtain a resonance free region for the resolvent of the Laplacian on $\Gamma\backslash \mathbb{H}^{d+1}$. Our approach is to construct a coding for the geodesic flow and then prove a Dolgopyat-type spectral estimate for the corresponding transfer operator

Protecting Privacy When Releasing Search Results from Medical Document Data

Health information technologies have greatly facilitated sharing of personal health data for secondary use, which is critical to medical and health research. However, there is a growing concern about privacy due to data sharing and publishing. Medical and health data typically contain unstructured text documents, such as clinical narratives, pathology reports, and discharge summaries. This study concerns privacy-preserving extraction, summary, and release of information from medical documents. Existing studies on privacy-preserving data mining and publishing focus mostly on structured data. We propose a novel approach to enable privacy-preserving extract, summarize, query and report patients’ demographic, health and medical information from medical documents. The extracted data is represented in a semi-structured, set-valued data format, which can be stored in a health information system for query and analysis. The privacy preserving mechanism is based on the cutting-edge idea of differential privacy, which offers rigorous privacy guarantee

Multi-period Optimal Control for Mobile Agents Considering State Unpredictability

The optimal control for mobile agents is an important and challenging issue. Recent work shows that using randomized mechanism in agents' control can make the state unpredictable, and thus improve the security of agents. However, the unpredictable design is only considered in single period, which can lead to intolerable control performance in long time horizon. This paper aims at the trade-off between the control performance and state unpredictability of mobile agents in long time horizon. Utilizing random perturbations consistent with uniform distributions to maximize the attackers' prediction errors of future states, we formulate the problem as a multi-period convex stochastic optimization problem and solve it through dynamic programming. Specifically, we design the optimal control strategy considering both unconstrained and input constrained systems. The analytical iterative expressions of the control are further provided. Simulation illustrates that the algorithm increases the prediction errors under Kalman filter while achieving the control performance requirements successfully

UFuzzer: Lightweight Detection of PHP-Based Unrestricted File Upload Vulnerabilities Via Static-Fuzzing Co-Analysis

Unrestricted file upload vulnerabilities enable attackers to upload malicious scripts to a web server for later execution. We have built a system, namely UFuzzer, to effectively and automatically detect such vulnerabilities in PHP-based server-side web programs. Different from existing detection methods that use either static program analysis or fuzzing, UFuzzer integrates both (i.e., static-fuzzing co-analysis). Specifically, it leverages static program analysis to generate executable code templates that compactly and effectively summarize the vulnerability-relevant semantics of a server-side web application. UFuzzer then “fuzzes” these templates in a local, native PHP runtime environment for vulnerability detection. Compared to static-analysis-based methods, UFuzzer preserves the semantics of an analyzed program more effectively, resulting in higher detection performance. Different from fuzzing-based methods, UFuzzer exercises each generated code template locally, thereby reducing the analysis overhead and meanwhile eliminating the need of operating web services. Experiments using real-world data have demonstrated that UFuzzer outperforms existing methods in either efficiency, or accuracy, or both. In addition, it has detected 31 unknown vulnerable PHP scripts including 5 CVEs