arXiv subjects

Yuru Zou

Publications and source records attributed to Yuru Zou.

13 recordsLinked to original sources

arXiv2026

Invariant measure for double base expansions

Given a pair $Q=(q_0,q_1)\in(1,\infty)^2$ with $q_0+q_1\ge q_0q_1$, a sequence $(c_i)\in\set{0,1}^\infty$ is called a $Q$-expansion of $x$ if \begin{equation*} x=\sum_{i=1}^{\infty}\frac{c_i}{q_{c_1}\cdots q_{c_i}}. \end{equation*} We primarily study the dynamical properties of the greedy and lazy maps, which are the piecewise-linear maps on the interval $I_Q=[0,\,1/(q_1-1)]$ defined by the corresponding algorithms for $Q$-expansions. We show that the greedy and lazy maps each of which has a unique absolutely continuous invariant probability measure, equivalent to the Lebesgue measure on the intervals \begin{equation*} \left[0,\frac{q_0}{q_1}\right)\qtq{and}\left(\frac{q_1}{q_0(q_1-1)}-1,\frac{1}{q_1-1}\right], \end{equation*} respectively. Furthermore, the corresponding dynamical systems are exact on $I_Q$. As a dynamical consequence, under the stronger condition $q_0+q_1>q_0q_1$ the set of points having unique $Q$-expansions has Lebesgue measure zero, and almost every $x\in I_{Q}$ admits a continuum of $Q$-expansions.

math.DS↗

arXiv2025

Hausdorff dimension of double base expansions and binary shifts with a hole

For two real bases $q_0, q_1 > 1$, a binary sequence $i_1 i_2 \cdots \in \{0,1\}^\infty$ is the $(q_0,q_1)$-expansion of the number \[ \pi_{q_0,q_1}(i_1 i_2 \cdots) = \sum_{k=1}^\infty \frac{i_k}{q_{i_1} \cdots q_{i_k}}. \] Let $U_{q_0,q_1}$ be the set of all real numbers having a unique $(q_0,q_1)$-expansion. When the bases are equal, i.e., $q_0 = q_1 = q$, Allaart and Kong (2019) established the continuity in $q$ of the Hausdorff dimension of the univoque set $U_{q,q}$, building on the work of Komornik, Kong, and Li (2017). We derive explicit formulas for the Hausdorff dimension of $U_{q_0,q_1}$ and the entropy of the underlying subshift for arbitrary $q_0, q_1 > 1$, and prove the continuity of these quantities as functions of $(q_0, q_1)$. Our results also concern general dynamical systems described by binary shifts with a hole, including, in particular, the doubling map with a hole and (linear) Lorenz maps.

math.DS↗

arXiv2025

Topology of univoque sets in double-base expansions

Given two real numbers $q_0,q_1>1$ satisfying $q_0+q_1\geq q_0q_1$ and two real numbers $d_0\ne d_1$, by a {double-base expansion} of a real number $x$ we mean a sequence $(i_k)\in \{0,1\}^{\infty}$ such that \begin{equation*} x=\sum_{k=1}^{\infty}\frac{d_{i_k}}{q_{{i_1}}q_{{i_2}}\cdots q_{{i_k}}}. \end{equation*} We denote by $\mathcal{U}_{{q_0,q_1}}$ the set of numbers $x$ having a unique expansion. The topological properties of $\mathcal{U}_{{q_0,q_1}}$ have been investigated in the equal-base case $q_0=q_1$ for a long time. We extend this research to the case $q_0\neq q_1$. While many results remain valid, a great number of new phenomena appear due to the increased complexity of double-base expansions.

math.DS↗

arXiv2024

Topological and dimensional properties of univoque bases in double-base expansions

Given two real numbers $q_0,q_1$ with $q_0, q_1 > 1$ satisfying $q_0+q_1 \ge q_0q_1$, we call a sequence $(d_i)$ with $d_i\in \{0,1\}$ a $(q_0,q_1)$-expansion or a double-base expansion of a real number $x$ if \[ x=\mathop{\sum}\limits_{i=1}^{\infty} \frac{d_{i}}{q_{d_1}q_{d_2}\cdots q_{d_i}}. \] When $q_0=q_1=q$, the set of univoque bases is given by the set of $q$'s such that $x = 1$ has exactly one $(q, q)$-expansion. The topological, dimensional and symbolic properties of such sets and their corresponding sequences have been intensively investigated. In our research, we study the topological and dimensional properties of the set of univoque bases for double-base expansions. This problem is more complicated, requiring new research strategies. Several new properties are uncovered. In particular, we show that the set of univoque bases in the double base setting is a meagre set with full Hausdorff dimension.

math.DS↗

arXiv2022

Unique double base expansions

For two real bases $q_0, q_1 > 1$, we consider expansions of real numbers of the form $\sum_{k=1}^{\infty} i_k/(q_{i_1}q_{i_2}\cdots q_{i_k})$ with $i_k \in \{0,1\}$, which we call $(q_0,q_1)$-expansions. A sequence $(i_k)$ is called a unique $(q_0,q_1)$-expansion if all other sequences have different values as $(q_0,q_1)$-expansions, and the set of unique $(q_0,q_1)$-expansions is denoted by $U_{q_0,q_1}$. In the special case $q_0 = q_1 = q$, the set $U_{q,q}$ is trivial if $q$ is below the golden ratio and uncountable if $q$ is above the Komornik--Loreti constant. The curve separating pairs of bases $(q_0, q_1)$ with trivial $U_{q_0,q_1}$ from those with non-trivial $U_{q_0,q_1}$ is the graph of a function $\mathcal{G}(q_0)$ that we call generalized golden ratio. Similarly, the curve separating pairs $(q_0, q_1)$ with countable $U_{q_0,q_1}$ from those with uncountable $U_{q_0,q_1}$ is the graph of a function $\mathcal{K}(q_0)$ that we call generalized Komornik--Loreti constant. We show that the two curves are symmetric in $q_0$ and $q_1$, that $\mathcal{G}$ and $\mathcal{K}$ are continuous, strictly decreasing, hence almost everywhere differentiable on $(1,\infty)$, and that the Hausdorff dimension of the set of $q_0$ satisfying $\mathcal{G}(q_0)=\mathcal{K}(q_0)$ is zero. We give formulas for $\mathcal{G}(q_0)$ and $\mathcal{K}(q_0)$ for all $q_0 > 1$, using characterizations of when a binary subshift avoiding a lexicographic interval is trivial, countable, uncountable with zero entropy and uncountable with positive entropy respectively. Our characterizations in terms of $S$-adic sequences including Sturmian and the Thue--Morse sequences are simpler than those of Labarca and Moreira (2006) and Glendinning and Sidorov (2015), and are relevant also for other open dynamical systems.

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arXiv2022

Learnable Irrelevant Modality Dropout for Multimodal Action Recognition on Modality-Specific Annotated Videos

With the assumption that a video dataset is multimodality annotated in which auditory and visual modalities both are labeled or class-relevant, current multimodal methods apply modality fusion or cross-modality attention. However, effectively leveraging the audio modality in vision-specific annotated videos for action recognition is of particular challenge. To tackle this challenge, we propose a novel audio-visual framework that effectively leverages the audio modality in any solely vision-specific annotated dataset. We adopt the language models (e.g., BERT) to build a semantic audio-video label dictionary (SAVLD) that maps each video label to its most K-relevant audio labels in which SAVLD serves as a bridge between audio and video datasets. Then, SAVLD along with a pretrained audio multi-label model are used to estimate the audio-visual modality relevance during the training phase. Accordingly, a novel learnable irrelevant modality dropout (IMD) is proposed to completely drop out the irrelevant audio modality and fuse only the relevant modalities. Moreover, we present a new two-stream video Transformer for efficiently modeling the visual modalities. Results on several vision-specific annotated datasets including Kinetics400 and UCF-101 validated our framework as it outperforms most relevant action recognition methods.

cs.CV↗

arXiv2021

EPSANet: An Efficient Pyramid Squeeze Attention Block on Convolutional Neural Network

Recently, it has been demonstrated that the performance of a deep convolutional neural network can be effectively improved by embedding an attention module into it. In this work, a novel lightweight and effective attention method named Pyramid Squeeze Attention (PSA) module is proposed. By replacing the 3x3 convolution with the PSA module in the bottleneck blocks of the ResNet, a novel representational block named Efficient Pyramid Squeeze Attention (EPSA) is obtained. The EPSA block can be easily added as a plug-and-play component into a well-established backbone network, and significant improvements on model performance can be achieved. Hence, a simple and efficient backbone architecture named EPSANet is developed in this work by stacking these ResNet-style EPSA blocks. Correspondingly, a stronger multi-scale representation ability can be offered by the proposed EPSANet for various computer vision tasks including but not limited to, image classification, object detection, instance segmentation, etc. Without bells and whistles, the performance of the proposed EPSANet outperforms most of the state-of-the-art channel attention methods. As compared to the SENet-50, the Top-1 accuracy is improved by 1.93% on ImageNet dataset, a larger margin of +2.7 box AP for object detection and an improvement of +1.7 mask AP for instance segmentation by using the Mask-RCNN on MS-COCO dataset are obtained. Our source code is available at:https://github.com/murufeng/EPSANet.

cs.CV↗

arXiv2021

Metric results for numbers with multiple $q$-expansions

Let $M$ be a positive integer and $q\in (1, M+1]$. A $q$-expansion of a real number $x$ is a sequence $(c_i)=c_1c_2\cdots$ with $c_i\in \{0,1,\ldots, M\}$ such that $x=\sum_{i=1}^{\infty}c_iq^{-i}$. In this paper we study the set $\mathcal{U}_q^j$ consisting of those real numbers having exactly $j$ $q$-expansions. Our main result is that for Lebesgue almost every $q\in (q_{KL}, M+1), $ we have $$\dim_{H}\mathcal{U}_{q}^{j}\leq \max\{0, 2\dim_H\mathcal{U}_q-1\}\text{ for all } j\in\{2,3,\ldots\}.$$ Here $q_{KL}$ is the Komornik-Loreti constant. As a corollary of this result, we show that for any $j\in\{2,3,\ldots\},$ the function mapping $q$ to $\dim_{H}\mathcal{U}_{q}^{j}$ is not continuous.

math.NT↗

arXiv2021

Expansions in multiple bases over general alphabets

Expansions in non-integer bases have been extensively investigated since a pioneering work of Rényi. We introduce a more general framework of alphabet-base systems that also includes Pedicini's general alphabets and the multiple-base expansions of Neunhäuserer and Li. We extend the Parry type lexicographic theory to this setup, and we improve and generalize various former results on unique expansions.

math.CO↗

arXiv2019

Univoque graphs and multiple expansions

Unique expansions in non-integer bases $q$ have been investigated in many papers during the last thirty years. They are often conveniently generated by labeled directed graphs. In the first part of this paper we give a precise description of the set of sequences generated by these graphs. Using the description of univoque graphs, the second part of the paper is devoted to the study of multiple expansions. Contrary to the unique expansions, we prove for each $j\ge 2$ that the set $U_q^j$ of numbers having exactly $j$ expansions is closed only if it is empty. Furthermore, generalizing an important example of Sidorov, we prove for a large class of bases that the Hausdorff dimension of $U_q^j$ is independent of $j$. In the last two sections our results are illustrated by many examples.

math.NT↗

arXiv2015

On a problem of countable expansions

For a real number $q\in(1,2)$ and $x\in[0,1/(q-1)]$, the infinite sequence $(d_i)$ is called a \emph{$q$-expansion} of $x$ if $$ x=\sum_{i=1}^\infty\frac{d_i}{q^i},\quad d_i\in\{0,1\}\quad\textrm{for all}~ i\ge 1. $$ For $m=1, 2, \cdots$ or $\aleph_0$ we denote by $\mathcal{B}_m$ the set of $q\in(1,2)$ such that there exists $x\in[0,1/(q-1)]$ having exactly $m$ different $q$-expansions. It was shown by Sidorov (2009) that $q_2:=\min \mathcal{B}_2\approx1.71064$, and later asked by Baker (2015) whether $q_2\in\mathcal{B}_{\aleph_0}$? In this paper we provide a negative answer to this question and conclude that $\mathcal{B}_{\aleph_0} $ is not a closed set. In particular, we give a complete description of $x\in[0,1/(q_2-1)]$ having exactly two different $q_2$-expansions.

math.NT↗

arXiv2015

Smallest bases of expansions with multiple digits

Given two positive integers $M$ and $k$, let $\B_k$ be the set of bases $q>1$ such that there exists a real number $x$ having exactly $k$ different $q$-expansions over the alphabet $\{0,1,\cdots,M\}$. In this paper we investigate the smallest base $q_2$ of $\B_2$, and show that if $M=2m$ the smallest base $$q_2 =\frac{m+1+\sqrt{m^2+2m+5}}{2},$$ and if $M=2m-1$ the smallest base $q_2$ is the appropriate root of $$ x^4=(m-1)\,x^3+2 m\, x^2+m \,x+1. $$ Moreover, for $M=2$ we show that $q_2$ is also the smallest base of $\B_k$ for all $k\ge 3$. This turns out to be different from that for $M=1$.

math.NT↗

arXiv2015

On small bases for which $1$ has countably many expansions

Let $q\in(1,2)$. A $q$-expansion of a number $x$ in $[0,\frac{1}{q-1}]$ is a sequence $(δ_i)_{i=1}^\infty\in\{0,1\}^{\mathbb{N}}$ satisfying $$ x=\sum_{i=1}^\infty\frac{δ_i}{q^i}.$$ Let $\mathcal{B}_{\aleph_0}$ denote the set of $q$ for which there exists $x$ with a countable number of $q$-expansions, and let $\mathcal{B}_{1, \aleph_0}$ denote the set of $q$ for which $1$ has a countable number of $q$-expansions. In \cite{Sidorov6} it was shown that $\min\mathcal{B}_{\aleph_0}=\min\mathcal{B}_{1,\aleph_0}=\frac{1+\sqrt{5}}{2},$ and in \cite{Baker} it was shown that $\mathcal{B}_{\aleph_0}\cap(\frac{1+\sqrt{5}}{2}, q_1]=\{ q_1\}$, where $q_1(\approx1.64541)$ is the positive root of $x^6-x^4-x^3-2x^2-x-1=0$. In this paper we show that the second smallest point of $\mathcal{B}_{1,\aleph_0}$ is $q_3(\approx1.68042)$, the positive root of $x^5-x^4-x^3-x+1=0$. Enroute to proving this result we show that $\mathcal{B}_{\aleph_0}\cap(q_1, q_3]=\{ q_2, q_3\}$, where $q_2(\approx1.65462)$ is the positive root of $x^6-2x^4-x^3-1=0$.

math.NT↗