arXiv subjects

Zhixiong Wen

Publications and source records attributed to Zhixiong Wen.

8 recordsLinked to original sources

arXiv2019

Sum-free sets generated by the period-k-folding sequences and some Sturmian sequences

First, we show that the sum-free set generated by the period-doubling sequence is not $κ$-regular for any $κ\geq 2$. Next, we introduce a generalization of the period-doubling sequence, which we call the period-$k$-folding sequences. We show that the sum-free sets generated by the period-$k$-folding sequences also fail to be $κ$-regular for all $κ\geq 2$. Finally, we study the sum-free sets generated by Sturmian sequences that begin with `11', and their difference sequences.

math.CO↗

arXiv2018

On the additive complexity of a Thue-Morse like sequence

In this paper, we study the additive complexity $ρ^{+}_{\mathbf{t}}(n)$ of a Thue-Morse like sequence $\mathbf{t}=σ^{\infty}(0)$ with the morphism $σ: 0\to 01, 1\to 12, 2\to 20$. We show that $ρ^{+}_{\mathbf{t}}(n)=2\lfloor\log_2(n)\rfloor+3$ for all integers $n\geq 1$. Consequently, $(ρ_{\mathbf{t}}(n))_{n\geq 1}$ is a $2$-regular sequence.

math.CO↗

arXiv2016

Morphisms on infinite alphabets, countable states automata and regular sequences

In this paper, we prove that a class of regular sequences can be viewed as projections of fixed points of uniform morphisms on a countable alphabet, and also can be generated by countable states automata. Moreover, we prove that the regularity of some regular sequences is invariant under some codings.

cs.FL↗

arXiv2016

On the abelian complexity of the Rudin-Shapiro sequence

In this paper, we study the abelian complexity of the Rudin-Shapiro sequence and a related sequence. We show that these two sequences share the same complexity function $ρ(n)$ which satisfies certain recurrence relations. As a consequence, the abelian complexity function is $2$-regular. Further, we prove that the box dimension of the graph of the asymptotic function $λ(x)$ is $3/2$ where $λ(x)=\lim_{k\to\infty}ρ(4^{k}x)/\sqrt{4^{k}x}$ and $ρ(x)=ρ(\lfloor x\rfloor)$ for any $x> 0$.

math.CO↗

arXiv2015

On the regularity of $\{\lfloor\log_b(αn+β)\rfloor\}_{n\geq0}$

Let $α,β$ be real numbers and $b\geq2$ be an integer. Allouche and Shallit showed that the sequence $\{\lfloorαn+β\rfloor\}_{n\geq0}$ is $b$-regular if and only if $α$ is rational. In this paper, using a base-independent regular language, we prove a similar result that the sequence $\{\lfloor\log_b(αn+β)\rfloor\}_{n\geq0}$ is $b$-regular if and only if $α$ is rational. In particular, when $α=\sqrt{2},β=0$ and $b=2$, we answer the question of Allouche and Shallit that the sequence $\{\lfloor\frac{1}{2}+\log_2n\rfloor\}_{n\geq0}$ is not $2$-regular, which has been proved by Bell, Moshe and Rowland respectively.

cs.FL↗

arXiv2015

Remarks on dimensions of Cartesian product sets

Given metric spaces $E$ and $F$, it is well known that $$\dim_HE+\dim_HF\leq\dim_H(E\times F)\leq\dim_HE+\dim_PF,$$ $$\dim_HE+\dim_PF\leq \dim_P(E\times F)\leq\dim_PE+\dim_PF,$$ and $$\underline{\dim}_BE+\overline{\dim}_BF \leq\overline{\dim}_B(E\times F) \leq\overline{\dim}_BE+\overline{\dim}_BF,$$ where $\dim_HE$, $\dim_PE$, $\underline{\dim}_BE$, $\overline{\dim}_BE$ denote the Hausdorff, packing, lower box-counting, and upper box-counting dimension of $E$, respectively. In this note we shall provide examples of compact sets showing that the dimension of the product $E\times F$ may attain any of the values permitted by the above inequalities. The proof will be based on a study on dimension of the product of sets defined by digit restrictions.

math.MG↗

arXiv2014

Doubling measures on uniform Cantor sets

We obtain a complete description for a probability measure to be doubling on an arbitrarily given uniform Cantor set. The question of which doubling measures on such a Cantor set can be extended to a doubling measure on [0; 1] is also considered.

math.MG↗

arXiv2013

On the irrationality exponent of the regular paperfolding numbers

In this paper, improving the method of Allouche \emph{et al.} \cite{APWW98}, we calculate the Hankel determinant of the regular paperfolding sequence, and prove that the Hankel determinant sequence module 2 is periodic with period 10 which answers Coon's conjecture \cite{CV12}. Then we extend Bugeaud's method \cite{Bugeaud11} to obatin the exact value of the irrationality exponent for some general transcendental numbers. Using the results above, we prove that the irrationality exponents of the regular paperfolding numbers are exactly 2.

math.NT↗