Roland Bacher
We define the category $Rec(\mathbb K)$ of recurrence matrices over a field $\mathbb K$ and use it to calculate determinants of Hankel matrices related to the Thue-Morse sequence.
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Nous décrivons quelques résultats récents sur la suite de Thue-Morse, ainsi que des questions ou conjectures, dont l'une, due à Shevelev, est résolue dans cet article. We describe some recent results on the Thue-Morse sequence. We also list open questions and conjectures, one of which is due to Shevelev and proved in this paper.
First we generalize the Thue-Morse sequence (the generalized Thue-Morse sequences) by a cyclic permutations and p-adic system, and consider the necessary-sufficient condition that it is non-periodic. Moreover if the generalized Thue-Morse sequence is not periodic, then all equally spaced subsequences of the generalized Thue-Morse sequences are not periodic. Finally we apply recently combinatorial transcendence results to the generalized non-periodic Thue-Morse sequence, we can found many transcendental numbers.
Let $(A_n)_{n\in \mathbb{N}}, (B_n)_{n\in \mathbb{N}} \in \mathbb{Z}^{\mathbb{N}}$ be two linear-recurrent sequences that meet a dominant root condition and a few more technical requirements. We show that the split family of Thue equations \[ |X(X-A_n Y)(X-B_n Y) - Y^3| = 1 \] has but the trivial solutions $\pm\{ (0,1), (1,0), (A_n,1), (B_n,1) \}$, if the parameter $n$ is larger than some effectively computable constant.
A novel generalization of the Prouhet-Thue-Morse sequence to binary $\pm 1$-weight sequences is presented. Derived from Rademacher functions, these weight sequences are shown to satisfy interesting orthogonality and recurrence relations. In addition, a result useful in describing these weight sequences as sidelobes of Doppler tolerant waveforms in radar is established.
For every $a\geq1$ we give a recursion algorithm of building of set of solutions of equations of the form $t(x+a)=t(x)$ and $t(x+a)=1-t(x),$ where $\{t(n)\}$ is Thue-Morse sequence. We pose an open problem and two conjectures.
In a recent paper, Baker and Kong have studied the Hausdorff dimension of the intersection of Cantor sets with their translations. We extend their results to more general Cantor sets. The proofs rely on the frequencies of digits in unique expansions in non-integer bases. In relation with this, we introduce a practical method to determine the frequency of any given finite block in Thue--Morse type sequences.
We revisit the relation between the von Koch curve and the Thue-Morse sequence given in a recent paper of Ma and Goldener by relating their study to papers written by Coquet and Dekking at the beginning of the 80s. We also emphasize that more general links between fractal objects and automatic sequences can be found in the literature.
On split families of Thue equations with linear recurrence sequences as factors
研究以线性递推序列为参数函数的 Thue 方程分裂族，证实了在参数大于某可计算常数时存在显式解集的结论对特定递推序列族同样成立。
We introduce and study two analogs of one of the best known sequence in Mathematics : Thue-Morse sequence. The first analog is concerned with the parity of number of runs of 1's in the binary representation of nonnegative integers. The second one is connected with the parity of number of 1's in the representation of nonnegative integers in so-called negabinary (or in base $-2).$ We give for them some recurrent and structure formulas and prove that the second $(0,1)$-sequence is cube-free, while the first one is quint-free. Finally we consider several interesting unsolved problems.
It is a classical problem in algebraic number theory to decide if a number field admits power integral bases and further to calculate all generators of power integral bases. This problem is especially delicate to consider in an infinite parametric family of number fields. In the present paper we investigate power integral bases in the infinite parametric family of pure quartic fields $K=Q(\sqrt[4]{m})$. We often pointed out close connection of various types of Thue equations with calculating power integral bases. In this paper we determine power integral bases in pure quartic fields by reducin…
In 1998, Allouche, Peyrière, Wen and Wen established that the Hankel determinants associated with the Thue--Morse sequence on $\{-1, 1\}$ are always nonzero. Their proof depends on a set of sixteen recurrence relations. We present an alternative, purely combinatorial proof of the same result. We also re-prove a recent result of Coons on the non-vanishing of the Hankel determinants associated to two other classical integer sequences.
On the absolute value of the autocorrelations of the Thue-Morse sequence
本文改进了 Baake 和 Coons 关于 Thue-Morse 序列自相关绝对值增长阶的结果，给出 $\alpha$ 的上下界，并揭示了两个 $k$-正则序列逐点乘积的线性表示结构。
We prove that the Thue--Morse sequence $\mathbf t$ along subsequences indexed by $\lfloor n^c\rfloor$ is normal, where $1<c<3/2$. That is, for $c$ in this range and for each $ω\in\{0,1\}^L$, where $L\geq 1$, the set of occurrences of $ω$ as a subword (contiguous finite subsequence) of the sequence $n\mapsto \mathbf t_{\lfloor n^c\rfloor}$ has asymptotic density $2^{-L}$. This is an improvement over a recent result by the second author, which handles the case $1<c<4/3$. In particular, this result shows that for $1<c<3/2$ the sequence $n\mapsto \mathbf t_{\lfloor n^c\rfloor}$ attains both of its…
Let $t_n = (-1)^{s_2(n)}$, where $s_2(n)$ is the sum of binary digits function. The sequence $(t_n)_{n\in \mathbb N}$ is the well-known Prouhet-Thue-Morse sequence. In this note we initiate the study of the sequence $(h_n)_{n\in \mathbb N}$, where $h_0 = 0, h_1 = $1 and for $n \ge 2$ we define $h_n$ recursively as follows:$ h_n = t_n h_{n-1} + h_{n-2}$. We prove several results concerning arithmetic properties of the sequence $(h_n )_{n\in \mathbb N}$. In particular, we prove non-vanishing of $h_n$ for $n \ge 5$, automaticity of the sequence $(h_n \pmod m)_{n\in \mathbb N}$ for each m, and oth…
Formal inverses of the generalized Thue-Morse sequences and variations of the Rudin-Shapiro sequence
本文研究广义Thue-Morse序列及Rudin-Shapiro序列变体的形式逆，给出其递推关系与自动机，并分析连续相同字母串长度和字母频率，与原始序列比较。
Let $s_{k}(n)$ denote the sum of digits of an integer $n$ in base $k$. Motivated by certain identities of Nieto, and Bateman and Bradley involving sums of the form $\sum_{i=0}^{2^{n}-1}(-1)^{s_{2}(i)}(x+i)^{m}$ for $m=n$ and $m=n+1$, we consider the sequence of polynomials \begin{equation*} f_{m,n}^{\mathbf u}(x)=\sum_{i=0}^{k^{n}-1}ζ_{k}^{s_{k}(i)}(x+{\mathbf u}(i))^{m}. \end{equation*} defined for sequences ${\bf u}(i)$ satisfying a certain recurrence relation. We prove that computing these polynomials is essentially equivalent with computing their constant term and we find an explicit formu…
Powers of the Thue--Morse Series: 2-Adic Valuations and Automatic Odd Parts
本文研究 Thue--Morse 生成函数幂的系数，证明特定指数族（$2^r$ 或 $3\cdot 2^r$）下奇数部分模 $2^s$ 的序列是 $2$-自动的，并给出 $m=6$ 时的赋值公式与自动性。
Linear independence of series related to the Thue--Morse sequence along powers
本文推广E. Miyanohara的结果，证明当β为大于√φ的Pisot数或Salem数时，与沿幂次的Thue–Morse序列相关的级数在ℚ(β)上线性无关，进而推出对应线性组合序列非最终周期。
Studying and comparing arithmetic properties of a given automatic sequence and the sequence of coefficients of the composition inverse of the associated formal power series (the formal inverse of that sequence) is an interesting problem. This problem was studied before for the Thue-Morse sequence. In this paper, we study arithmetic properties of the formal inverses of two sequences closely related to the well-known Baum-Sweet sequence. We give the recurrence relations for their formal inverses and we determine whether the sequences of indices at which these formal inverses take value $0$ and $…
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