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Yao-Qiang Li

Publications and source records attributed to Yao-Qiang Li.

10 recordsLinked to original sources
arXiv2025

Spectrality and supports of infinite convolutions in $\mathbb{R}^d$

We study the spectrality of a class of infinite convolutions in $\mathbb{R}^d$, generalizing a result given by Li, Miao and Wang in 2022 from $\mathbb{R}$ to $\mathbb{R}^d$. This allows us to easily construct spectral measures with and without compact supports in $\mathbb{R}^d$, and motivates us to systematically study the supports of infinite convolutions. In particular, we give a sufficient and necessary condition for infinite convolutions to exist with compact supports, generalizing a related well-known result which is widely used. After giving strong relations between supports of infinite convolutions and sets of infinite sums, we study the closedness and fractal dimensions of infinite sums of union sets in order to deal with non-compact supports of infinite convolutions. As an application of these new tools, we deduce that there are spectral measures with and without compact supports of arbitrary Hausdorff and packing dimensions in $\mathbb{R}^d$, generalizing another result given by Li, Miao and Wang in 2022 from $\mathbb{R}$ to $\mathbb{R}^d$.

math.FA↗
arXiv2021

Hausdorff dimension of frequency sets in beta-expansions

By applying a 2014 result on the distribution of full cylinders, we give a proof of the useful folklore: for any $β>1$, the Hausdorff dimension of an arbitrary set in the shift space $S_β$ is equal to the Hausdorff dimension of its natural projection in $[0,1]$. It has been used in some former papers without proof. Then we clarify that for calculating the Hausdorff dimension of frequency sets using variational formulae, one only needs to focus on the Markov measures of explicit order when the $β$-expansion of $1$ is finite. Concretely, it suffices to optimize a function with finitely many variables under some restrictions. Finally, as an application, we obtain an exact formula for the Hausdorff dimension of frequency sets for an important class of $β$'s, which are called pseudo-golden ratios (also called multinacci numbers).

math.DS↗
arXiv2020

Generalized Koch curves and Thue-Morse sequences

Let $(t_n)_{n\ge0}$ be the well konwn $\pm1$ Thue-Morse sequence $$+1,-1,-1,+1,-1,+1,+1,-1,\cdots.$$ Since the 1982-1983 work of Coquet and Dekking, it is known that $\sum_{k<n}t_ke^\frac{2kπi}{3}$ is strongly related to the famous Koch curve. As a natural generalization, for integer $m\ge1$, we use $\sum_{k<n}δ_ke^\frac{2kπi}{m}$ to define generalized Koch curve, where $(δ_n)_{n\ge0}$ is the generalized Thue-Morse sequence defined to be the unique fixed point of the morphism $$+1\mapsto+1,+δ_1,\cdots,+δ_m$$ $$-1\mapsto-1,-δ_1,\cdots,-δ_m$$ beginning with $δ_0=+1$ and $δ_1,\cdots,δ_m\in\{+1,-1\}$, and we prove that generalized Koch curves are the attractors of corresponding iterated function systems. For the case that $m\ge2$, $δ_0=\cdots=δ_{\lfloor\frac{m}{4}\rfloor}=+1$, $δ_{\lfloor\frac{m}{4}\rfloor+1}=\cdots=δ_{m-\lfloor\frac{m}{4}\rfloor-1}=-1$ and $δ_{m-\lfloor\frac{m}{4}\rfloor}=\cdots=δ_m=+1$, the open set condition holds, and then the corresponding generalized Koch curve has Hausdorff, packing and box dimension $\log(m+1)/\log|\sum_{k=0}^mδ_ke^{\frac{2kπi}{m}}|$, where taking $m=3$ and then $δ_0=+1,δ_1=δ_2=-1,δ_3=+1$ will recover the result on the classical Koch curve.

math.DS↗
arXiv2020

Expansions in multiple bases

Expansion of real numbers is a basic research topic in number theory. Usually we expand real numbers in one given base. In this paper, we begin to systematically study expansions in multiple given bases in a reasonable way, which is a generalization in the sense that if all the bases are taken to be the same, we return to the classical expansions in one base. In particular, we focus on greedy, quasi-greedy, lazy, quasi-lazy and unique expansions in multiple bases.

math.DS↗
arXiv2020

Divisibility properties of factors of the discriminant of generalized Fibonacci numbers

We study some divisibility properties related to the factors of the discriminant of the characteristic polynomial of generalized Fibonacci sequences $(G_n)_{n\ge0}$ defined by $G_0=0$, $G_1=1$ and $G_n=pG_{n-1}+qG_{n-2}$ for $n\ge2$, where $p,q$ are given integers. As corollaries, we give some divisibility properties on some well known sequences.

math.NT↗
arXiv2020

Infinite products related to generalized Thue-Morse sequences

Given an integer $q\ge2$ and $θ_1,\cdots,θ_{q-1}\in\{0,1\}$, let $(θ_n)_{n\ge0}$ be the generalized Thue-Morse sequence, defined to be the unique fixed point of the morphism $$0\mapsto0θ_1\cdotsθ_{q-1}$$ $$1\mapsto1\overlineθ_1\cdots\overlineθ_{q-1}$$ beginning with $θ_0:=0$, where $\overline{0}:=1$ and $\overline{1}:=0$. For rational functions $R$, we study infinite products of the forms $$\prod_{n=1}^\infty\Big(R(n)\Big)^{(-1)^{θ_n}}\quad\text{and}\quad\prod_{n=1}^\infty\Big(R(n)\Big)^{θ_n}.$$ This generalizes relevant results given by Allouche, Riasat and Shallit in 2019 on infinite products related to the famous Thue-Morse sequence $(t_n)_{n\ge0}$ of the forms $$\prod_{n=1}^\infty\Big(R(n)\Big)^{(-1)^{t_n}}\quad\text{and}\quad\prod_{n=1}^\infty\Big(R(n)\Big)^{t_n}.$$

math.NT↗
arXiv2020

Hausdorff dimension of frequency sets of univoque sequences

For integer $m\ge3$, we study the dynamical system $(Λ_m,σ_m)$ where $Λ_m$ is the set $\{w\in\{0,1\}^\mathbb{N}: w$ does not contain $0^m$ or $1^m\}$ and $σ_m$ is the shift map on $\{0,1\}^\mathbb{N}$ restricted to $Λ_m$, study the Bernoulli-type measures on $Λ_m$ and find out the unique equivalent $σ_m$-invariant ergodic probability measure. As an application, we obtain the Hausdorff dimension of the set of univoque sequences, the Hausdorff dimension of the set of sequences in which the lengths of consecutive $0$'s and consecutive $1$'s are bounded, and the Hausdorff dimension of their frequency subsets.

math.DS↗
arXiv2019

Random walks associated to beta-shifts

We study the dynamics of a simple random walk on subshifts defined by the beta transformation and apply it to find concrete formulae for the Hausdorff dimension of digit frequency sets for $β>1$ that solves $β^{m+1}-β^m-1=0$ generalising the work of Fan and Zhu. We also give examples of $β$ where this approach fails.

math.DS↗
arXiv2019

Digit frequencies of beta-expansions

Let $β>1$ be a non-integer. First we show that Lebesgue almost every number has a $β$-expansion of a given frequency if and only if Lebesgue almost every number has infinitely many $β$-expansions of the same given frequency. Then we deduce that Lebesgue almost every number has infinitely many balanced $β$-expansions, where an infinite sequence on the finite alphabet $\{0,1,\cdots,m\}$ is called balanced if the frequency of the digit $k$ is equal to the frequency of the digit $m-k$ for all $k\in\{0,1,\cdots,m\}$. Finally we consider variable frequency and prove that for every pseudo-golden ratio $β\in(1,2)$, there exists a constant $c=c(β)>0$ such that for any $p\in[\frac{1}{2}-c,\frac{1}{2}+c]$, Lebesgue almost every $x$ has infinitely many $β$-expansions with frequency of zeros equal to $p$.

math.DS↗