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Baowei Wang

Publications and source records attributed to Baowei Wang.

16 recordsLinked to original sources
arXiv2026

Hausdorff dimension of sets of numbers whose continued fractions contain arbitrarily long arithmetic progressions

Continued fractions with prescribed structures on sequences of their partial quotients have been intensively studied in the literature. As far as an integer sequence, especially a randomly generated one is concerned, an attractive question is whether it contains arbitrarily long arithmetic progressions. In this paper we study the fractal structure of irrational numbers whose sequences of partial quotients are strictly increasing and contain arbitrarily long, quantified arithmetic progressions.

math.NT↗
arXiv2026

NiMark: A Non-intrusive Watermarking Framework against Screen-shooting Attacks

Unauthorized screen-shooting poses a critical data leakage risk. Resisting screen-shooting attacks typically requires high-strength watermark embedding, inevitably degrading the cover image. To resolve the robustness-fidelity conflict, non-intrusive watermarking has emerged as a solution by constructing logical verification keys without altering the original content. However, existing non-intrusive schemes lack the capacity to withstand screen-shooting noise. While deep learning offers a potential remedy, we observe that directly applying it leads to a previously underexplored failure mode, the Structural Shortcut: networks tend to learn trivial identity mappings and neglect the image-watermark binding. Furthermore, even when logical binding is enforced, standard training strategies cannot fully bridge the noise gap, yielding suboptimal robustness against physical distortions. In this paper, we propose NiMark, an end-to-end framework addressing these challenges. First, to eliminate the structural shortcut, we introduce the Sigmoid-Gated XOR (SG-XOR) estimator to enable gradient propagation for the logical operation, effectively enforcing rigid image-watermark binding. Second, to overcome the robustness bottleneck, we devise a two-stage training strategy integrating a restorer to bridge the domain gap caused by screen-shooting noise. Experiments demonstrate that NiMark consistently outperforms representative state-of-the-art methods against both digital attacks and screen-shooting noise, while maintaining zero visual distortion.

eess.IV↗
arXiv2025

Hausdorff measures of sets in Exact Diophantine approximation

Let $(X, d)$ be a compact metric space, and let $Q \subset X$ be countable. Given functions $R: Q \to \mathbb{R}^+$ and $\phi: \mathbb{R}^+ \to \mathbb{R}^+$, we consider the set $E(Q, R, \phi)$ of points $x \in X$ that ``hit'' the shrinking balls $B({\xi},{\phi(R(\xi))})$ for infinitely many $\xi \in Q$, yet, for every $\epsilon \in (0,1)$, are eventually ``cleared out'' from the slightly smaller neighborhoods $B({\xi},{(1-\epsilon)\phi(R(\xi))})$, that is, they lie outside all but finitely many of these smaller balls. We give sufficient conditions (also necessary under mild assumptions) for $E(Q, R, \phi)$ to have infinite Hausdorff $f$-measure. This setting generalizes both the classical set $\mathrm{Exact}(\psi)$ of exactly $\psi$-approximable points (with $\psi$ non-increasing) and certain types of restricted Diophantine approximation sets.

math.NT↗
arXiv2025

Log-Hausdorff multifractality of the absolutely continuous spectral measure of the almost Mathieu operator

This paper focuses on the fractal characteristics of the absolutely continuous spectral measure of the subcritical almost Mathieu operator (AMO) and Diophantine frequency. In particular, we give a complete description of the (classical) multifractal spectrum and a finer description in the logarithmic gauge. The proof combines continued$-$fraction$/$metric Diophantine techniques and refined covering arguments. These results rigorously substantiate (and quantify in a refined gauge) the physicists' intuition that the absolutely continuous component of the spectrum is dominated by energies with trivial scaling index, while also exhibiting nontrivial exceptional sets which are negligible for classical Hausdorff measure but large at the logarithmic scale.

math-ph↗
arXiv2025

Sim-to-Real: An Unsupervised Noise Layer for Screen-Camera Watermarking Robustness

Unauthorized screen capturing and dissemination pose severe security threats such as data leakage and information theft. Several studies propose robust watermarking methods to track the copyright of Screen-Camera (SC) images, facilitating post-hoc certification against infringement. These techniques typically employ heuristic mathematical modeling or supervised neural network fitting as the noise layer, to enhance watermarking robustness against SC. However, both strategies cannot fundamentally achieve an effective approximation of SC noise. Mathematical simulation suffers from biased approximations due to the incomplete decomposition of the noise and the absence of interdependence among the noise components. Supervised networks require paired data to train the noise-fitting model, and it is difficult for the model to learn all the features of the noise. To address the above issues, we propose Simulation-to-Real (S2R). Specifically, an unsupervised noise layer employs unpaired data to learn the discrepancy between the modeled simulated noise distribution and the real-world SC noise distribution, rather than directly learning the mapping from sharp images to real-world images. Learning this transformation from simulation to reality is inherently simpler, as it primarily involves bridging the gap in noise distributions, instead of the complex task of reconstructing fine-grained image details. Extensive experimental results validate the efficacy of the proposed method, demonstrating superior watermark robustness and generalization compared to state-of-the-art methods.

cs.CV↗
arXiv2023

The Shrinking Target Problem for Matrix Transformations of Tori: revisiting the standard problem

Let $T$ be a $d\times d$ matrix with real coefficients. Then $T$ determines a self-map of the $d$-dimensional torus ${\Bbb T}^d={\mathbb{R}}^d/{\Bbb Z}^d$. Let $ \{E_n \}_{n \in \mathbb{N}} $ be a sequence of subsets of ${\Bbb T}^d$ and let $W(T,\{E_n \})$ be the set of points $\mathbf{x} \in {\Bbb T}^d$ such that $T^n(\mathbf{x})\in E_n $ for infinitely many $n\in {\mathbb{N}}$. For a large class of subsets (namely, those satisfying the so called bounded property $ ({\boldsymbol{\rm B}}) $ which includes balls, rectangles, and hyperboloids) we show that the $d$-dimensional Lebesgue measure of the shrinking target set $W(T,\{E_n \})$ is zero (resp. one) if a natural volume sum converges (resp. diverges). In fact, we prove a quantitative form of this zero-one criteria that describes the asymptotic behaviour of the counting function $R(x,N):= \# \big\{ 1\le n \le N : T^{n}(x) \in E_n \} $. The counting result makes use of a general quantitative statement that holds for a large class measure-preserving dynamical systems (namely, those satisfying the so called summable-mixing property). We next turn our attention to the Hausdorff dimension of $W(T,\{E_n \})$. In the case the subsets $E_n$ are balls, rectangles or hyperboloids we obtain precise formulae for the dimension. These shapes correspond, respectively, to the simultaneous, weighted and multiplicative theories of classical Diophantine approximation. The dimension results for balls generalises those obtained in an earlier paper by Hill and the third-named author for integer matrices to real matrices. In the final section, we discuss various problems that stem from the results proved in the paper.

math.NT↗
arXiv2021

Mass transference principle from rectangles to rectangles in Diophantine approximation

By introducing a ubiquity property for rectangles, we prove the mass transference principle from rectangles to rectangles, i.e., if a sequence of rectangles forms a ubiquity system (a full measure property), then the limsup set defined by shrinking these rectangles to smaller rectangles has full Hausdorff measure or to say transfer a full measure property to a full Hausdorff measure property for brevity. The limsup sets generated by balls or generated by rectangles appear at the most fundamental level in Diophantine approximation: one follows from Dirichlet's theorem, the other follows from Minkowski's theorem. So the result sets up a general principle for the Hausdorff measure theory for high dimensional Diophantine approximation which, together with the landmark work of Beresnevich & Velani in 2006 where a transference principle from balls to balls is established, gives a coherent Hausdorff measure theory for metric Diophantine approximation. The dimensional theory for limsup sets generated by rectangles also underpins the dimensional theory in multiplicative Diophantine approximation where unexpected phenomenon occurs and the usually used methods or even their generalizations fail to work.

math.NT↗
arXiv2021

Mahler's question for intrinsic Diophantine approximation on triadic Cantor set: the divergence theory

In this paper, we consider the intrinsic Diophantine approximation on the triadic Cantor set $\mathcal{K}$, i.e. approximating the points in $\mathcal{K}$ by rational numbers inside $\mathcal{K}$, a question posed by K. Mahler. By using another height function of a rational number in $\mathcal{K}$, i.e. the denominator obtained from its periodic 3-adic expansion, a complete metric theory for this variant intrinsic Diophantine approximation is presented which yields the divergence theory of Mahler's original question.

math.NT↗
arXiv2021

Diophantine analysis of the expansions of a fixed point under continuum many bases

In this paper, we study the Diophantine properties of the orbits of a fixed point in its expansions under continuum many bases. More precisely, let $T_β$ be the beta-transformation with base $β>1$, $\{x_{n}\}_{n\geq 1}$ be a sequence of real numbers in $[0,1]$ and $φ\colon \mathbb{N}\rightarrow (0,1]$ be a positive function. With a detailed analysis on the distribution of {\em full cylinders} in the base space $\{β>1\}$, it is shown that for any given $x\in(0,1]$, for almost all or almost no bases $β>1$, the orbit of $x$ under $T_β$ can $φ$-well approximate the sequence $\{x_{n}\}_{n\geq 1}$ according to the divergence or convergence of the series $\sum φ(n)$. This strengthens Schmeling's result significantly and complete all known results in this aspect. Moreover, the idea presented here can also be used to determine the Lebesgue measure of the set \begin{equation*} \{x\in [0,1]\colon|T^{n}_βx-L(x)|<φ(n) \text{ for infinitely many } n\in\mathbb{N}\}, \end{equation*} for a fixed base $β>1$, where $L\colon [0,1]\rightarrow[0,1]$ is a Lipschitz function.

math.NT↗
arXiv2020

Dynamical Borel-Cantelli lemma for recurrence theory

We study the dynamical Borel-Cantelli lemma for recurrence sets in a measure preserving dynamical system $(X, μ, T)$ with a compatible metric $d$. We prove that, under some regularity conditions, the $μ$-measure of the following set \[ R(ψ)= \{x\in X : d(T^n x, x) < ψ(n)\ \text{for infinitely many}\ n\in\N \} \] obeys a zero-full law according to the convergence or divergence of a certain series, where $ψ:\N\to\R^+$. Some of the applications of our main theorem include the continued fractions dynamical systems, the beta dynamical systems, and the homogeneous self-similar sets.

math.DS↗
arXiv2018

Multifractal analysis of the Birkhoff sums of Saint-Petersburg potential

Let $((0,1], T)$ be the doubling map in the unit interval and $φ$ be the Saint-Petersburg potential, defined by $φ(x)=2^n$ if $x\in (2^{-n-1}, 2^{-n}]$ for all $n\geq 0$. We consider the asymptotic properties of the Birkhoff sum $S\_n(x)=φ(x)+\cdots+φ(T^{n-1}(x))$. With respect to the Lebesgue measure, the Saint-Petersburg potential is not integrable and it is known that $\frac{1}{n\log n}S\_n(x)$ converges to $\frac{1}{\log 2}$ in probability. We determine the Hausdorff dimension of the level set $\{x: \lim\_{n\to\infty}S\_n(x)/n=α\} \ (α>0)$, as well as that of the set $\{x: \lim\_{n\to\infty}S\_n(x)/Ψ(n)=α\} \ (α>0)$, when $Ψ(n)=n\log n, n^a $ or $2^{n^γ}$ for $a>1$, $γ>0$. The fast increasing Birkhoff sum of the potential function $x\mapsto 1/x$ is also studied.

math.DS↗
arXiv2017

Hausdorff dimension of the set approximated by irrational rotations

Let $θ$ be an irrational number and $φ: {\mathbb N} \to {\mathbb R}^{+}$ be a monotone decreasing function tending to zero. Let $$E_φ(θ) =\Big\{y \in \mathbb R: \|nθ- y\|<φ(n), \ {\text{for infinitely many}}\ n\in {\mathbb N} \Big\}, $$ i.e. the set of points which are approximated by the irrational rotation with respect to the error function $φ(n)$. In this article, we give a complete description of the Hausdorff dimension of $E_φ(θ)$ for any monotone function $φ$ and any irrational $θ$.

math.NT↗
arXiv2013

Quantitative recurrence properties in conformal iterated function systems

Let $Λ$ be a countable index set and $S=\{ϕ_i: i\in Λ\}$ be a conformal iterated function system on $[0,1]^d$ satisfying the open set condition. Denote by $J$ the attractor of $S$. With each sequence $(w_1,w_2,...)\in Λ^{\mathbb{N}}$ is associated a unique point $x\in [0,1]^d$. Let $J^\ast$ denote the set of points of $J$ with unique coding, and define the mapping $T:J^\ast \to J^\ast$ by $Tx= T (w_1,w_2, w_3...) = (w_2,w_3,...)$. In this paper, we consider the quantitative recurrence properties related to the dynamical system $(J^\ast, T)$. More precisely, let $f:[0,1]^d\to \mathbb{R}^+$ be a positive function and $$R(f):=\{x\in J^\ast: |T^nx-x|<e^{-S_n f(x)}, \ {\text{for infinitely many}}\ n\in \mathbb{N}\},$$ where $S_n f(x)$ is the $n$th Birkhoff sum associated with the potential $f$. In other words, $R(f)$ contains the points $x$ whose orbits return close to $x$ infinitely often, with a rate varying along time. Under some conditions, we prove that the Hausdorff dimension of $R(f)$ is given by $\inf\{s\ge 0: P(T, -s(f+\log |T'|))\le 0\}$, where $P$ is the pressure function and $T'$ is the derivative of $T$. We present some applications of the main theorem to Diophantine approximation.

math.DS↗
arXiv2013

Diophantine approximation of the orbit of 1 in the dynamical system of bete expansions

We consider the distribution of the orbits of the number 1 under the $β$-transformations $T_β$ as $β$ varies. Mainly, the size of the set of $β>1$ for which a given point can be well approximated by the orbit of 1 is measured by its Hausdorff dimension. That is, the dimension of the following set $$ E\big({\ell_n}_{n\ge 1}, x_0\big)=\Big{β>1: |T^n_β1-x_0|<β^{-\ell_n}, {for infinitely many} n\in \N\Big} $$ is determined, where $x_0$ is a given point in $[0,1]$ and ${\ell_n}_{n\ge 1}$ is a sequence of integers tending to infinity as $n\to \infty$. For the proof of this result, the notion of the recurrence time of a word in symbolic space is introduced to characterize the lengths and the distribution of cylinders (the set of $β$ with a common prefix in the expansion of 1) in the parameter space ${β\in \R: β>1}$.

math.DS↗