2.1 Morphic sequences

Consider an alphabet $\Sigma = \{0,\ldots ,k-1\}$ . We endow the set $\Sigma ^{\omega }$ of infinite words over $\Sigma $ with the product topology, where $\Sigma $ has the discrete topology. The shift map $\sigma :\Sigma ^{\omega }\rightarrow \Sigma ^{\omega }$ is defined by $\sigma (u_0u_1u_2 \cdots ) = u_1u_2u_3\cdots $ .

A morphism is a homomorphism $\varphi : \Sigma ^+ \rightarrow \Sigma ^+$ of the free semigroup $\Sigma ^+$ . The incidence matrix $M_\varphi \in \mathbb N^{k\times k}$ of $\varphi $ is defined by taking $(M_\varphi )_{i,j}$ to be the number of occurrences of the symbol j in $\varphi (i)$ . We say that $\varphi $ is primitive if some power of $M_\varphi $ is positive and we say that $\varphi $ has exponential growth if the spectral radius $\rho (M_{\varphi })$ of $M_\varphi $ is strictly greater than one. If $M_\varphi $ is primitive then, by the Perron–Frobenius theorem, its spectral radius is a strictly positive real eigenvalue and all other eigenvalues have absolute value strictly less than $\rho (M_{\varphi })$ . We say that $\varphi $ is irreducible if its characteristic polynomial is irreducible over $\mathbb Q$ . We define $\varphi $ to be of Pisot type (or simply Pisot) if $\rho (M_{\varphi })$ is a Pisot number, that is, all its Galois conjugates have absolute value strictly less than one.

Let $\varphi $ be a primitive morphism over alphabet $\Sigma $ . Assume that $\varphi $ is prolongable, that is, $\varphi (0) = 0u$ for some word $u \in \Sigma ^+$ . Then, the sequence $(\varphi ^n(0))_{n\geq 0}$ of finite words converges to an infinite word $\boldsymbol u$ that is a fixed point of $\varphi $ . The word $\boldsymbol u$ is uniformly recurrent, that is, each finite factor occurs infinitely often and there is an upper bound on the length of the gaps between every pair of successive occurrences of the factor.

We refer to the shift dynamical system $(X_\varphi ,\sigma )$ , where $X_\varphi $ is the topological closure of the orbit of $\boldsymbol u$ under the shift map and, by a slight abuse of notation, $\sigma $ denotes the restriction of the shift map to $X_\varphi $ . For a primitive morphism, this dynamical system has a unique ergodic measure $\mu $ and we have a unitary operator $U_\varphi : L^2(X_\varphi ,\mu ) \rightarrow L^2(X_\varphi ,\mu )$ , defined by $U_\varphi (f)=f\circ \sigma $ for f a square-integrable function on the measure space $(X_\varphi ,\mu )$ . The morphism $\varphi $ is said to have pure discrete spectrum if the space $L^2(X_\varphi ,\mu )$ has a basis of eigenvectors of $U_\varphi $ .

2.2 Number theory

Let K be a number field of degree d over $\mathbb Q$ and let $M(K)$ be the set of places of K. We divide $M(K)$ into the collection of Archimedean places, which are determined either by an embedding of K in $\mathbb {R}$ or a complex-conjugate pair of embeddings of K in $\mathbb {C}$ , and the set of non-Archimedean places, which are determined by prime ideals in the ring $\mathcal {O}_K$ of integers of K.

For $a \in K$ and $v \in M(K)$ , define the absolute value $|a|_v$ as follows: $|a|_v := |\sigma (a)|^{1/d}$ if v corresponds to a real embedding $\sigma :K\rightarrow \mathbb {R}$ ; $|a|_v := |\sigma (a)|^{2/d}$ if v corresponds to a complex-conjugate pair of embeddings $\sigma ,\overline {\sigma }:K \rightarrow \mathbb {C}$ ; $|a|_v := N(\mathfrak {p})^{-\mathrm {ord}_{\mathfrak {p}}(a)/d}$ if v corresponds to a prime ideal $\mathfrak {p}$ in $\mathcal {O}$ and $\mathrm {ord}_{\mathfrak {p}}(a)$ is the order to which $\mathfrak {p}$ divides the ideal $a\mathcal {O}$ . With the above definitions, we have the product formula: $\prod _{v \in M(K)} |a|_v = 1$ for all $a \in K^\times $ . Given a set of places $S\subseteq M(K)$ , the ring $\mathcal {O}_S$ of S-integers is the subring comprising all $a \in K$ such that $|a|_v \leq 1$ for all non-Archimedean places $v\not \in S$ .

For $m\geq 1$ , the Weil height of the projective point $\boldsymbol {a}=[a_0 : a_1 : \cdots : a_m] \in \mathbb {P}^m(K)$ is

$$ \begin{align*} H(\boldsymbol{a}):=\prod_{v \in M(K)}\max(|a_0|_v,\ldots,|a_m|_v) \, .\end{align*} $$

This definition is independent of the choice of the field K containing $a_0,\ldots ,a_m$ . We define the height $H(a)$ of $a \in K$ to be the height $H([1 : a])$ of the corresponding point in $\mathbb {P}^1(K)$ . For a non-zero Laurent polynomial $f = x^n \sum _{i=0}^m a_i x^i \in K[x,x^{-1}]$ , where $m\geq 1$ and $n\in \mathbb Z$ , following [Reference Lenstra20], we define its height $H(f)$ to be the height $H([a_0 : \cdots : a_m])$ of the vector of coefficients.

The following special case of the p-adic subspace theorem of Schlickewei [Reference Schlickewei28] is one of the main ingredients of our approach. (We formulate the special case of the subspace theorem in which all but one of the linear forms are coordinate variables.)

Theorem 2.1. Let $S \subseteq M(K)$ be a finite set of places of K that contains all Archimedean places. Let $v_0 \in S$ be a distinguished place and choose a continuation of $|\cdot |_{v_0}$ to $\overline {\mathbb Q}$ , also denoted $|\cdot |_{v_0}$ . Given $m\geq 2$ , let $L(x_1,\ldots ,x_{m})$ be a linear form with algebraic coefficients and let $i_0 \in \{1,\ldots ,m\}$ be a distinguished index such that $x_{i_0}$ has non-zero coefficient in $\overline {\mathbb Q}$ . Then, for any $\varepsilon>0$ , the set of solutions $\boldsymbol {a}=(a_1,\ldots ,a_m) \in (\mathcal {O}_S)^m$ of the inequality

$$ \begin{align*} |L(\boldsymbol{a})|_{v_0}\cdot \bigg( \prod_{\substack{(i,v)\in \{1,\ldots,m\}\times S\\(i,v) \neq (i_0,v_0)}}|a_i|_v \bigg) \leq H(\boldsymbol{a})^{-\varepsilon} \end{align*} $$

is contained in a finite union of proper linear subspaces of $K^m$ .

We need the following proposition about roots of univariate polynomials.

We also need the following separation bound [Reference Waldschmidt30, Ch. 3].

Proposition 2.2. Let $f \in \mathbb Z[x]$ have degree d and let L be the sum of the absolute value of its coefficients. If $\beta \in \overline {\mathbb Q}$ is not a root of f, then

$$ \begin{align*} |f(\beta)|> \frac{1}{c^d L^{\mathrm{deg}(\beta)}} \, , \end{align*} $$

where c is a constant that depends only on the height of $\beta $ .

3.1 The Fibonacci word

Let $\Sigma =\{0,1\}$ and consider the morphism $\varphi :\Sigma ^+\rightarrow \Sigma ^+$ given by $\varphi (0)=01$ and $\varphi (1)=0$ . The Fibonacci word $\boldsymbol u_{\mathrm {Fib}} \in \Sigma ^\omega $ is the morphic word

$$ \begin{align*} \boldsymbol u_{\mathrm{Fib}} := \lim_{n\rightarrow \infty}\varphi^n(0) = \mathtt{01001010010010100} \cdots \,. \end{align*} $$

The Fibonacci word is not periodic and, hence, $\boldsymbol u_{\mathrm {Fib}}$ is not equal to any of its tails $\sigma ^n(\boldsymbol u_{\mathrm {Fib}})$ for $n>0$ . However, if the shift n is well chosen, then the mismatches between $\boldsymbol u_{\mathrm {Fib}}$ and $\sigma ^n(\boldsymbol u_{\mathrm {Fib}})$ are sparse. This intuition will be formalised in the definition of echoing word. A good choice of shifts are the elements of the sequence $\langle 1,2,3,5,8,\ldots \rangle $ of Fibonacci numbers: for example, vertically aligning $\boldsymbol u_{\mathrm {Fib}}$ and $\sigma ^5(\boldsymbol u_{\mathrm {Fib}})$ , and shading mismatches, we have

Observe that each mismatch involves a factor 10 of $\boldsymbol u_{\mathrm {Fib}}$ for which the corresponding factor in $\sigma ^5(\boldsymbol u_{\mathrm {Fib}})$ is the reverse, 01. In fact, we see the same phenomenon for all shifts of $\boldsymbol u_{\mathrm {Fib}}$ by an element of the Fibonacci sequence. Furthermore, it turns out that for each successive shift, the distance between the mismatched factors increases. This is formalised below as the expanding gaps property.

3.2 The Tribonacci word

The following example illustrates some more complicated phenomena than the Fibonacci case. Let $\Sigma =\{0,1,2\}$ and consider the morphism $\varphi :\Sigma ^+\rightarrow \Sigma ^+$ given by $\varphi (0)=01$ , $\varphi (1)=02$ and $\varphi (2)=0$ . The Tribonacci word $\boldsymbol u_{\mathrm {Tri}} \in \Sigma ^\omega $ is the morphic word

$$ \begin{align*} \boldsymbol u_{\mathrm{Tri}} := \lim_{n\rightarrow \infty}\varphi^n(0) = \mathtt{0102010010201} \cdots \, .\end{align*} $$

Associated with the Tribonacci word, we have the sequence $\langle t_n \rangle _{n=0}^\infty $ of Tribonacci numbers, defined by the recurrence $t_n = t_{n-1}+t_{n-2}+t_{n-3}$ and initial conditions $t_0=1, t_1=2,t_2=4$ . It is easy to see that the word $\varphi ^n(0)$ has length $t_n$ for all $n\in \mathbb {N}$ .

In the spirit of our analysis of the Fibonacci word, we match the Tribonacci word against shifts of itself by elements of the Tribonacci sequence $\langle 1,2,4,7,13,24,\ldots \rangle $ . For example, comparing $\boldsymbol u_{\mathrm {Tri}}$ and $\sigma ^{13}(\boldsymbol u_{\mathrm {Tri}})$ , we have

Notice that the shaded mismatches appear as a fixed set of factors (either 10, 20 or 102) in $\boldsymbol u_{\mathrm {Tri}}$ that get reversed in $ \sigma ^{13}(\boldsymbol u_{\mathrm {Tri}})$ . Unlike with the Fibonacci word, this time, the factors may appear close to each other. Nevertheless, by suitably grouping neighbouring mismatch factors, we recover a form of the expanding gaps property, and we are moreover able to show that the mismatches between $\boldsymbol u_{\mathrm {Tri}}$ and its shifts are relatively sparse.

3.3 Definition of echoing words

Drawing on the respective examples of the Fibonacci and Tribonacci words, we give in this section the formal definition of echoing word. To this end, we introduce the following notation and terminology. Given two non-empty intervals $I,J \subseteq \mathbb {N}$ , write $I<J$ if $a<b$ for all $a\in I$ and $b\in J$ , and define the distance of I and J to be $d(I,J):=\min \{|a-b|:a\in I,b\in J\}$ . Furthermore, define the density of a non-empty finite set $S\subseteq \mathbb N$ to be $\mathrm {den}(S):={|S|}/{\max (S)}$ . For two functions $f(n)$ and $g(n)$ , we write $f\ll g$ if there exist positive constants $n_0,c_1$ such that for all $n\geq n_0$ , we have $f(n) \leq ~c_1 g(n)$ . We also write $f\asymp g$ if both $f\ll g$ and $g\ll f$ .

The notion of echoing word concerns the alignment of the suffixes $\sigma ^{r_n}(\boldsymbol u)$ and $\sigma ^{s_n}(\boldsymbol u)$ of $\boldsymbol u$ , where $s_n-r_n$ tends to infinity. The conditions imply that the set of discrepancies between these two suffixes is contained in a union of intervals of asymptotic density at most $\varepsilon $ and such that the distance between successive intervals grows linearly with $s_n$ .

We show that the Fibonacci and Tribonacci words are both echoing. As suggested by the examples above, in the case of the Fibonacci word, a suitable choice of the sequence $\langle s_n\rangle _{n=0}^\infty $ of shifts is the sequence of Fibonacci numbers, while in the case of the Tribonacci word, it is the sequence of Tribonacci numbers (and in both cases, $\langle r_n\rangle _{n=0}^\infty $ is the all-zero sequence). In the case of the Fibonacci word, the intervals $I_{n,j}$ are doubletons, whereas in the case of the Tribonacci word, their total length grows with n, although the asymptotic density of $\bigcup _{j=1}^\infty I_{n,j}$ tends to zero as n tends to infinity.

4.1 Balanced pairs

A pair of words $(x,y) \in \Sigma ^+\times \Sigma ^+$ is said to be balanced if x and y have the same commutative image. A balanced pair is said to be irreducible if it cannot be decomposed as a product (in the semigroup $\Sigma ^+\times \Sigma ^+$ ) of two or more balanced pairs. For $x \in \Sigma $ , we call the balanced pair $(x,x)$ a coincidence. An irreducible balanced pair that is not a coincidence is called a mismatch.

Let $\varphi :\Sigma ^+ \rightarrow \Sigma ^+$ be a morphism. Note that if $(x,y)$ is a balanced pair, then so is $(\varphi (x),\varphi (y))$ . A finite set $\Gamma \subseteq \Sigma ^+\times \Sigma ^+$ of irreducible balanced pairs is said to be closed for $\varphi $ if for all $(x,y) \in \Gamma $ , we can write $(\varphi (x),\varphi (y))$ as a product of balanced pairs in $\Gamma $ . If $\Gamma $ is closed, then $\varphi $ naturally lifts to a morphism $\widetilde {\varphi }$ on $\Gamma $ such that $\widetilde {\varphi }((x,y))$ is the decomposition of $(\varphi (x),\varphi (y))$ as a product of elements of $\Gamma $ . We note that $\widetilde {\varphi }$ necessarily maps a coincidence to a string of coincidences. We say that $\widetilde {\varphi }$ satisfies the coincidence condition if for all $(x,y) \in \Gamma $ , there exists $k\geq 0$ such that the string $\widetilde {\varphi }^k((x,y))$ contains a coincidence. It is not difficult to see that if $\widetilde {\varphi }$ satisfies the coincidence condition, then the asymptotic upper density of mismatch symbols in $\widetilde {\varphi }^k((x,y))$ decays to zero at an exponential rate in k.

Let $\boldsymbol u$ be a fixed point of a primitive morphism $\varphi $ . Given a non-empty prefix x of $\boldsymbol u$ , since $\boldsymbol u$ is recurrent, we can write $\boldsymbol u = x y x \boldsymbol u'$ for some $y \in \Sigma ^*$ and suffix $\boldsymbol u' \in \Sigma ^{\omega }$ . Given as input the prefix x, the balanced-pair algorithm [Reference Livshits21, Reference Livshits22] successively computes all irreducible factors of the balanced pairs $(\varphi ^k(xy),\varphi ^k(yx))$ for $k=1,2,\ldots .$ We say that the algorithm terminates with coincidence if the resulting set $\Gamma $ of irreducible factors is finite and the lifting of $\varphi $ to a morphism $\widetilde {\varphi }$ on $\Gamma $ satisfies the coincidence condition.

The following result is [Reference Akiyama, Barge, Berthé, Lee, Siegel, Kellendonk, Lenz and Savinien6, Theorem 5.3]. The fact that $(X_\varphi ,\sigma )$ has pure discrete spectrum if the balanced-pair algorithm for $\varphi $ terminates with coincidence on some input prefix is due to Livshits [Reference Livshits21, Reference Livshits22]. The converse direction follows by combining [Reference Sirvent and Solomyak29, Corollary 5.3] and [Reference Clark and Sadun10, Corollary 3.2].

Hollander and Solomyak [Reference Hollander and Solomyak16] showed that the balanced-pair algorithm terminates for irreducible Pisot morphisms on two-element alphabets, while Barge and Diamond [Reference Barge and Diamond8] showed that the output satisfies the coincidence condition. This gives the following theorem.

4.2 The Tribonacci word is echoing

This section presents an extended example of an echoing word. The construction is only sketched below, and will be treated more formally and in more generality in Lemma 4.4 and Theorem 7.1.

Consider the alphabet $\Sigma =\{0,1,2\}$ and the Tribonacci morphism $\varphi : \Sigma ^+\rightarrow \Sigma ^+$ , defined by $\varphi (0)=01$ , $\varphi (1)=02$ and $\varphi (2)=0$ . We introduce the alphabet $\Gamma := \{a_0,\ldots ,a_{10}\}$ of irreducible balanced pairs, whose elements (depicted as tiles) are as follows:

$$ \begin{align*} \begin{array}{llllll}a_0= \left[\begin{array}{@{}c@{\;}c@{}}0&1\\1&0\end{array}\right] &a_1= \left[\begin{array}{@{}c@{\;}c@{}}1&0\\0&1\end{array}\right] &a_2= \left[\begin{array}{@{}c@{\;}c@{}}0&2\\2&0\end{array}\right] &a_3= \left[\begin{array}{@{}c@{\;}c@{}}2&0\\0&2\end{array}\right] &a_4= \left[\begin{array}{@{}c@{\;}c@{\;}c@{}}1&0&2\\2&0&1\end{array}\right] &a_5= \left[\begin{array}{@{}c@{\;}c@{\;}c@{}}2&0&1\\1&0&2\end{array}\right]\\[5pt]a_6= \left[\begin{array}{@{}c@{\;}c@{\;}c@{\;}c@{}}0&1&0&2\\2&0&1&0\end{array}\right] &a_7= \left[\begin{array}{@{}c@{\;}c@{\;}c@{\;}c@{}}2&0&1&0\\0&1&0&2\end{array}\right] &a_8= \left[\begin{array}{@{}c@{}}0\\0\end{array}\right] &a_9= \left[\begin{array}{@{}c@{}}1\\1\end{array}\right] &a_{10}= \left[\begin{array}{@{}c@{}}2\\2\end{array}\right]\,. &\end{array} \end{align*} $$

We partition $\Gamma $ into the subset $\Gamma _0:=\{a_0,\ldots ,a_7\}$ of mismatches and the subset $\Gamma _1:=\{a_8,a_9,a_{10}\}$ of coincidences.

The map $\varphi $ lifts to a morphism $\widetilde {\varphi } : \Gamma ^+\rightarrow \Gamma ^+$ that is defined as follows:

$$ \begin{align*} \begin{array}{r@{}c@{}lr@{}c@{}l}\widetilde{\varphi}\bigg( \left[\begin{array}{@{}c@{\;}c@{}}0&1\\1&0\end{array}\right] \bigg)& \, := \, & \left[\begin{array}{@{}c@{}}0\\0\end{array}\right] \left[\begin{array}{@{}c@{\;}c@{\;}c@{}}1&0&2\\2&0&1\end{array}\right]&\widetilde{\varphi}\bigg( \left[\begin{array}{@{}c@{\;}c@{\;}c@{}}1&0&2\\2&0&1\end{array}\right] \bigg)&:=& \left[\begin{array}{@{}c@{}}0\\0\end{array}\right] \left[\begin{array}{@{}c@{\;}c@{\;}c@{\;}c@{}}2&0&1&0\\0&1&0&2\end{array}\right]\\\widetilde{\varphi}\bigg( \left[\begin{array}{@{}c@{\;}c@{}}1&0\\0&1\end{array}\right] \bigg)&\, :=\, & \left[\begin{array}{@{}c@{}}0\\0\end{array}\right] \left[\begin{array}{@{}c@{\;}c@{\;}c@{}}2&0&1\\1&0&2\end{array}\right]&\widetilde{\varphi}\bigg( \left[\begin{array}{@{}c@{\;}c@{\;}c@{}}2&0&1\\1&0&2\end{array}\right] \bigg)&:=& \left[\begin{array}{@{}c@{}}0\\0\end{array}\right] \left[\begin{array}{@{}c@{\;}c@{\;}c@{\;}c@{}}0&1&0&2\\2&0&1&0\end{array}\right]\\\widetilde{\varphi}\bigg( \left[\begin{array}{@{}c@{\;}c@{}}0&2\\2&0\end{array}\right] \bigg)&:=& \left[\begin{array}{@{}c@{}}0\\0\end{array}\right] \left[\begin{array}{@{}c@{\;}c@{}}1&0\\0&1\end{array}\right]&\widetilde{\varphi}\bigg( \left[\begin{array}{@{}c@{\;}c@{\;}c@{\;}c@{}}0&1&0&2\\2&0&1&0\end{array}\right] \bigg)&:=& \left[\begin{array}{@{}c@{}}0\\0\end{array}\right] \left[\begin{array}{@{}c@{\;}c@{}}1&0\\0&1\end{array}\right] \left[\begin{array}{@{}c@{\;}c@{}}2&0\\0&2\end{array}\right] \left[\begin{array}{@{}c@{\;}c@{}}1&0\\0&1\end{array}\right]\\\widetilde{\varphi}\bigg( \left[\begin{array}{@{}c@{\;}c@{}}2&0\\0&2\end{array}\right] \bigg)&:=& \left[\begin{array}{@{}c@{}}0\\0\end{array}\right] \left[\begin{array}{@{}c@{\;}c@{}}0&1\\1&0\end{array}\right]&\widetilde{\varphi}\bigg( \left[\begin{array}{@{}c@{\;}c@{\;}c@{\;}c@{}}2&0&1&0\\0&1&0&2\end{array}\right] \bigg)&:=& \left[\begin{array}{@{}c@{}}0\\0\end{array}\right] \left[\begin{array}{@{}c@{\;}c@{}}0&1\\1&0\end{array}\right] \left[\begin{array}{@{}c@{\;}c@{}}0&2\\2&0\end{array}\right] \left[\begin{array}{@{}c@{\;}c@{}}0&1\\1&0\end{array}\right]\end{array} \end{align*} $$

and

$$ \begin{align*} \widetilde{\varphi}\bigg( \left[\begin{array}{@{}c@{}}0\\0\end{array}\right] \bigg):= \left[\begin{array}{@{}c@{}}0\\0\end{array}\right] \left[\begin{array}{@{}c@{}}1\\1\end{array}\right]\quad\qquad\widetilde{\varphi}\bigg( \left[\begin{array}{@{}c@{}}1\\1\end{array}\right] \bigg):= \left[\begin{array}{@{}c@{}}0\\0\end{array}\right] \left[\begin{array}{@{}c@{}}2\\2\end{array}\right]\quad\qquad\widetilde{\varphi}\bigg( \left[\begin{array}{@{}c@{}}2\\2\end{array}\right] \bigg):= \left[\begin{array}{@{}c@{}}0\\0\end{array}\right] \, . \end{align*} $$

By inspection, $\widetilde {\varphi }$ satisfies the coincidence condition.

Let the projection morphisms $\pi _1,\pi _2 :\Gamma ^+ \rightarrow \Sigma ^+$ be defined by $\pi _1(x,y):=x$ and $\pi _2(x,y):=y$ for all $(x,y)\in \Gamma $ . Denote by $\boldsymbol u \in \Sigma ^{\omega }$ the Tribonacci word and by

$$ \begin{align*} \boldsymbol w = \left[\begin{array}{@{}c@{\;}c@{}}0&1\\1&0\end{array}\right] \left[\begin{array}{@{}c@{\;}c@{}}0&2\\2&0\end{array}\right] \left[\begin{array}{@{}c@{\;}c@{}}0&1\\1&0\end{array}\right] \left[\begin{array}{@{}c@{}}0\\0\end{array}\right] \left[\begin{array}{@{}c@{\;}c@{}}0&1\\1&0\end{array}\right] \left[\begin{array}{@{}c@{\;}c@{}}0&2\\2&0\end{array}\right] \left[\begin{array}{@{}c@{\;}c@{}}0&1\\1&0\end{array}\right] \left[\begin{array}{@{}c@{\;}c@{}}0&1\\1&0\end{array}\right]\cdots \in \Gamma^{\omega} \end{align*} $$

the image of $\boldsymbol u$ under the coding $\iota :\Sigma ^+\rightarrow \Gamma ^+$ given by

$$ \begin{align*} \iota(0) := \left[\begin{array}{@{}c@{\;}c@{}}0&1\\1&0\end{array}\right]\quad\iota(1) := \left[\begin{array}{@{}c@{\;}c@{}}0&2\\2&0\end{array}\right]\quad\iota(2) := \left[\begin{array}{@{}c@{}}0\\0\end{array}\right] \, . \end{align*} $$

Then, $\pi _1(\boldsymbol w) = \boldsymbol u$ and $\pi _2(\boldsymbol w)=\sigma (\boldsymbol u)$ . Furthermore, for $n_0 \in \mathbb N$ , we can write

(4.1) $$ \begin{align} \widetilde{\varphi}^{n_0}(\boldsymbol w) = w_0 a_{i_1} {w}_{1} a_{i_2} w_{2} a_{i_3} \cdots\,, \end{align} $$

where $ {w}_0, {w}_1,\ldots \in \Gamma ^*$ are sequences of coincidence symbols and $a_{i_1},a_{i_2},\ldots \in \Gamma $ are mismatch symbols. Since $\widetilde {\varphi }$ satisfies the coincidence condition, by making $n_0\in \mathbb {N}$ sufficiently large, the upper asymptotic density of mismatch symbols in $\widetilde {\varphi }^{n_0}(\boldsymbol {w})$ can be made arbitrarily small.

The data to show the echoing property (see Definition 3.1) are as follows. For all $n\in \mathbb {N}$ , consider the image of (4.1) under the n-fold application of $\widetilde {\varphi }$ :

(4.2) $$ \begin{align} \widetilde{\varphi}^{n+n_0}(\boldsymbol{w}) = \widetilde{\varphi}^{n}(w_0) \underbrace{\widetilde{\varphi}^{n}(a_{i_1})}_{I_{n,1}} \widetilde{\varphi}^{n}(w_1) \underbrace{\widetilde{\varphi}^{n}(a_{i_2})}_{I_{n,2}} \widetilde{\varphi}^{n}(w_2) \underbrace{\widetilde{\varphi}^{n}(a_{i_3})}_{I_{n,3}} \cdots \,. \end{align} $$

We define $r_n:=0$ and $s_n:=|\varphi ^{n+n_0}(0)|$ . We also have $\pi _1(\widetilde {\varphi }^{n+n_0}(\boldsymbol w)) =\boldsymbol u$ and $\pi _2(\widetilde {\varphi }^{n+n_0}(\boldsymbol w)) =\sigma ^{s_n}(\boldsymbol u)$ . For all $j\in \{1,2,\ldots \}$ , we define $I_{n,j} \subseteq \mathbb {N}$ to be the smallest interval of positions in $\boldsymbol u=\pi _1(\widetilde {\varphi }^{n+n_0}(\boldsymbol w))$ that contains all mismatch symbols occurring in the factor $\pi _1(\widetilde {\varphi }^{n}(a_{i_j}))$ (see (4.2) for an illustration). By construction, this choice satisfies the covering property. Intuitively, the density property follows from the fact that, by primitivity of $\varphi $ , for all $n\in \mathbb N$ , the upper asymptotic density of $\bigcup _{j=1}^\infty I_{n,j}$ is bounded above by a constant multiple of the upper asymptotic density of mismatch symbols in $\widetilde {\varphi }^{n_0}(\boldsymbol {w})$ , which can be made arbitrarily small as previously noted.

It remains to consider the expanding gaps property. Here, we note that, by definition of $r_n$ and $s_n$ , we clearly have $\lim _{n\rightarrow \infty }(s_n-r_n)=\infty $ . Moreover, the primitivity of $\varphi $ can be used to show that $d(I_{n,j},I_{n,j+1})\gg s_n$ for all $j\in \mathbb N$ . Note that this bound holds even when the factor $w_j$ in (4.1) is empty, since $\widetilde \varphi (a_{i_{j+1}})$ starts with a coincidence symbol.

4.3 Pisot morphic words are echoing

The following is the main result of this section.

Theorem 4.3 follows from Theorems 4.1 and 4.2 (which respectively give conditions for the balanced-pair algorithm to terminate with coincidence over binary and general alphabets) and the following lemma (showing that termination with coincidence implies the echoing condition). The proof of Lemma 4.4 generalises the construction for the Tribonacci word in §4.2.

Proof. By assumption, there exists a finite alphabet $\Gamma \subseteq \Sigma ^+\times \Sigma ^+$ of irreducible balanced pairs and a lifting of $\varphi $ to a morphism $\widetilde {\varphi } : \Gamma ^+ \rightarrow \Gamma ^+$ satisfying the coincidence condition. More specifically, we suppose that $\Gamma $ is the output of the balanced-pair algorithm on input x a non-empty prefix of $\boldsymbol u$ . Since x occurs in $\boldsymbol u$ with bounded gaps, we may write $\boldsymbol u = x y_1 x y_2 x \cdots $ for some $y_1,y_2,\ldots \in \Sigma ^*$ and obtain an infinite word $\boldsymbol w \in \Gamma ^{\omega }$ by decomposing the infinite sequence of balanced pairs

(4.3) $$ \begin{align} \begin{bmatrix} x y_1\\ y_1 x \end{bmatrix} \begin{bmatrix} x y_2 \\ y_2 x \end{bmatrix} \begin{bmatrix} x y_3 \\ y_3 x \end{bmatrix} \cdots \end{align} $$

into a sequence of irreducible balanced pairs in $\Gamma $ . Note that $\pi _1(\boldsymbol w) = \boldsymbol u$ and $\pi _2(\boldsymbol w) = \sigma ^{\ell }(\boldsymbol u)$ , where $\ell =|x|$ . Since $\varphi $ is primitive, there exists $c_0>0$ such that for all $\gamma ,\gamma ' \in \Gamma $ and all $n\in \mathbb N$ , we have

(4.4) $$ \begin{align} |\pi_1(\widetilde{\varphi}^n(\gamma))| \leq c_0\, |\pi_1(\widetilde{\varphi}^n(\gamma'))| \,. \end{align} $$

Assume that $\boldsymbol u$ is not ultimately periodic. Then, given $n_0 \in \mathbb {N}$ , the word $\widetilde {\varphi }^{n_0}(\boldsymbol w)$ contains infinitely many mismatch symbols. Moreover, since $\widetilde {\varphi }$ satisfies the coincidence condition, for $n_0$ sufficiently large, $\widetilde {\varphi }^{n_0}(\boldsymbol w)$ also contains infinitely many coincidence symbols. For such $n_0$ , we thus have either

(4.5) $$ \begin{align}\widetilde{\varphi}^{n_0}(\boldsymbol w)= { z}_0 { v}_0 { z}_1 { v_1} { z}_2 \cdots \quad\text{or}\quad \widetilde{\varphi}^{n_0}(\boldsymbol w)= { v}_0 { z}_0 { v}_1 { z_1} { v}_2 \cdots\,, \end{align} $$

where ${ z}_0,{ z}_1,\ldots \in \Gamma ^+$ are non-null sequences of mismatch symbols and ${ v}_0,{ v}_1,\ldots \in \Gamma ^+$ are non-null sequences of coincidence symbols. Let $\varepsilon>0$ be given as in the definition of an echoing sequence. Since $\widetilde {\varphi }$ satisfies the coincidence condition, for sufficiently large $n_0\in \mathbb {N}$ , the upper asymptotic density of mismatch symbols in $\widetilde {\varphi }^{n_0}(\boldsymbol {w})$ is at most $\varepsilon /c_0$ .

The data to show the echoing property (see Definition 3.1) are as follows. Let us assume the left-hand case in (4.5); the reasoning for the right-hand case follows with minor changes. For all $n\in \mathbb {N}$ , consider the string

(4.6) $$ \begin{align} \widetilde{\varphi}^{n+n_0}(\boldsymbol{w}) = \widetilde{\varphi}^{n}( z_0) \widetilde{\varphi}^{n}( v_0) \underbrace{ \widetilde{\varphi}^{n}({ z}_1)}_{I_{n,1}} \widetilde{\varphi}^{n}( v_1) \underbrace{ \widetilde{\varphi}^{n}({ z}_2)}_{I_{n,2}} \widetilde{\varphi}^{n}( v_2) \cdots \,. \end{align} $$

Then, for all $n\in \mathbb N$ , we have $\pi _1(\widetilde {\varphi }^{n+n_0}(\boldsymbol w)) = \boldsymbol u$ and $\pi _2(\widetilde {\varphi }^{n+n_0}(\boldsymbol w)) = \sigma ^{t_n}(\boldsymbol u)$ , where $t_n:= |\varphi ^{n+n_0}(x)|$ .

For all $n\in \mathbb N$ , define $r_n:=|\pi _1(\widetilde {\varphi }^n( z_0))|$ and $s_n:=r_n+t_n$ . For all $j\in \{1,2,\ldots \}$ , we define $I_{n,j} \subseteq \mathbb {N}$ to be the interval of positions in $\boldsymbol u =\pi _1(\widetilde {\varphi }^{n+n_0}(\boldsymbol w))$ corresponding to the factor $\pi _1(\widetilde {\varphi }^{n}(z_j))$ (see (4.6)). By construction, this choice satisfies the covering property. Since the mismatch symbols in $\widetilde {\varphi }^{n_0}(\boldsymbol w)$ have upper asymptotic density at most $\varepsilon /c_0$ , by (4.4), we have $\mathrm {den}(\bigcup _{j=1}^m I_{n,j})\leq \varepsilon $ for m sufficiently large; hence, the density property holds. Finally, by primitivity of $\varphi $ , $\pi _1(\widetilde \varphi ^n(\gamma ))$ has the same asymptotic growth rate as a function of n for all $\gamma \in \Gamma $ . It follows that $d(I_{n,j},I_{n,j+1})\gg s_n$ for all $j\in \mathbb N$ , with the implied constant being independent of j. Since $s_n-r_n=|\varphi ^{n+n_0}(x)|$ , we clearly have $\lim _{n\rightarrow \infty }(s_n-r_n)=\infty $ . This establishes the expanding gaps property.

6.1 The non-vanishing condition

The following definition presents a strengthening of the echoing condition that allows proving transcendence outright (removing the possibility that $[\![ \boldsymbol {u}]\!] _{\beta }$ be algebraic). The change to Definition 3.1 involves strengthening the expanding-gaps condition (to specify both lower and upper bounds on the gaps between intervals) and adding a fourth condition, called non-vanishing. The latter guarantees the existence of at least two intervals of mismatches that are non-zero when evaluated in base $\beta $ .

In §7, we show that the substitutive sequence defined by the k-Bonacci morphism is strongly echoing for all $k\geq 2$ . The case $k=2$ (the Fibonacci word) is straightforward. In the following section, we treat the case $k=3$ (the Tribonacci word), continuing the extended example from §4.2.

6.2 The Tribonacci word is strongly echoing

For $\Sigma :=\{0,1,2\}$ , let $\varphi : \Sigma ^+\rightarrow \Sigma ^+$ be the morphism that defines the Tribonacci word $\boldsymbol u \in \Sigma ^\omega $ and let $\widetilde {\varphi } : \Gamma ^+\rightarrow \Gamma ^+$ be its lifting to the closed set of balanced pairs, as defined in §4.2, with $\pi _1,\pi _2:\Gamma ^+\rightarrow \Sigma ^+$ the associated projection morphisms. We recall the construction showing that $\boldsymbol u$ is echoing and argue that the non-vanishing condition is also satisfied. In particular, we refer to the shifts $r_n:=0$ and $s_n:=|\varphi ^{n+n_0}(0)|$ , and intervals $I_{n,j}$ that witness that $\boldsymbol u$ is echoing.

For all $n\in \mathbb N$ and $j\in \{1,2,\ldots \}$ , define the polynomial $P_{n,j}(x) \in \mathbb {Z}[x]$ by

(6.1) $$ \begin{align} P_{n,j}(x) := \sum_{i \in I_{n,j}} (u_{i+s_n}-u_{i+r_n})x^{i} \,. \end{align} $$

Inspecting condition (4) of Definition 6.1, to show that $\boldsymbol u$ is non-vanishing, we must prove that for all $\beta $ such that $|\beta |>1$ , there exist indices $j_0<j_1$ such that $P_{n,j_0}(\beta ^{-1})\neq 0$ and $P_{n,j_1}(\beta ^{-1})\neq 0$ for infinitely many n. To this end, we give a recursive characterisation of the polynomials $P_{n,j}$ .

Recall the mismatch symbols $a_0,\ldots ,a_7$ in $\Gamma $ . For all $i \in \{0,\ldots ,7\}$ and $n\in \mathbb {N}$ , define the polynomial $Q_{n,i}(x) \in \mathbb Z[x]$ by the equation

(6.2) $$ \begin{align} Q_{n,i}(x):=\sum_{k=0}^{\ell-1} (\pi_1(\widetilde{\varphi}^n(a_i))_k - \pi_2(\widetilde{\varphi}^n(a_i))_k)x^{k} \, , \end{align} $$

where $\ell =|\pi _1(\widetilde {\varphi }(a_i))|$ . For example, we have $Q_{0,6}(x) = -2+x-x^2+2x^3$ since $a_6 = \big[\begin{smallmatrix}0&1&0&2\\ 2&0&1&0\end{smallmatrix}\big]$ . Referring to the list of symbols $a_{i_1},a_{i_2},\ldots $ and the corresponding intervals $I_{n,j}$ in (4.2), we observe that for all $n\in \mathbb N$ and $j \in \{1,2,\ldots \}$ , polynomial $P_{n,j}$ is the product $Q_{n,i_j}$ and a monomial. Thus, our task is to find $j_0<j_1$ such that $Q_{n,i_{j_0}}(\beta ^{-1})\neq 0$ and $Q_{n,i_{j_1}}(\beta ^{-1})\neq 0$ for infinitely many n.

We claim that for all n, there exists $i\in \{0,\ldots ,7\}$ such that $Q_{n,i}(\beta ^{-1}) \neq 0$ . From the claim, it follows that there exists $i^* \in \{0,\ldots ,7\}$ such that $Q_{n,i^*}(\beta ^{-1}) \neq 0$ for infinitely many $n \in \mathbb N$ . Since $\widetilde {\varphi }$ acts primitively on the set of mismatch symbols, for $n_0$ sufficiently large, ${i^*}$ occurs infinitely often in the list $i_1,i_2,\ldots $ in (4.1). In particular, there exist $j_0<j_1$ such that $Q_{n,i_{j_0}}(\beta ^{-1})\neq 0$ and $Q_{n,i_{j_1}}(\beta ^{-1})\neq 0$ for infinitely many n. This is what we wanted to prove and it remains to justify the claim.

For all $n\in \mathbb N$ , we have the vector $\boldsymbol {Q}_n:=(Q_{n,0},\ldots ,Q_{n,7}) \in \mathbb {Z}[x]^{8}$ of polynomials. We want to show that $\boldsymbol {Q}_n(\beta ^{-1})\neq 0$ for all n. The action of $\widetilde {\varphi }$ gives rise to a recurrence for $\boldsymbol Q_n$ . For example, we have $\widetilde {\varphi }(a_6) = a_8a_1a_3a_1$ and, hence,

$$ \begin{align*} \widetilde{\varphi}^{n+1}(a_6) = \widetilde{\varphi} ^n(a_8) \widetilde{\varphi} ^n(a_1) \widetilde{\varphi} ^n(a_3) \widetilde{\varphi} ^n(a_1) \,. \end{align*} $$

We deduce that

$$ \begin{align*}Q_{n+1,6}(x)= x^{\ell_{n,8}}Q_{n,1}(x)+ x^{\ell_{n,8}+\ell_{n,1}} Q_{n,3}(x)+ x^{\ell_{n,8}+\ell_{n,1}+\ell_{n,2}}Q_{n,1}(x)\, ,\end{align*} $$

where $\ell _{n,i} = |\pi _1(\widetilde {\varphi }^n(a_i))|$ for all $i\in \{0,\ldots ,8\}$ . Calculating the corresponding recurrences for each polynomial $Q_{n+1,i}(x)$ yields the vector recurrence $\boldsymbol {Q}_{n+1} = M_n \, \boldsymbol {Q}_n$ , where

(6.3) $$ \begin{align} M_n := x^{\ell_{n,8}} \cdot \begin{pmatrix} 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 &1+x^{\ell_{n,1}+\ell_{n,3}} & 0 & x^{\ell_{n,1}} & 0 & 0 & 0 & 0 \\ 1+x^{\ell_{n,0}+\ell_{n,2}} & 0 & x^{\ell_{n,0}} & 0 & 0& 0 & 0 & 0 \end{pmatrix}. \end{align} $$

However, $Q_{0,0}=(x-1)$ and $\mathrm {det}(M_n)=x^{\ell _{n,0}+\ell _{n,1}+8\ell _{n,8}}$ . Hence, for all $\beta \neq 1$ , we have $\boldsymbol {Q}_0(\beta ^{-1}) \neq \boldsymbol {0}$ and, by induction on n, $\boldsymbol {Q}_n(\beta ^{-1}) \neq \boldsymbol {0}$ for all $n\in \mathbb N$ . The claim is proven and this completes the argument that $\boldsymbol u$ is strongly echoing.

6.3 Transcendence result

For a strongly echoing sequence $\boldsymbol u$ and algebraic $\beta $ , we can show that $[\![ \boldsymbol {u}]\!] _{\beta }$ is outright transcendental. Whereas in the proof of Theorem 5.1 we assume that $[\![ \boldsymbol {u}]\!] _{\beta }$ is algebraic and use the subspace theorem to show that it lies in $\mathbb Q(\beta )$ , in the following, we continue the argument by applying the subspace theorem a second time (employing the non-vanishing condition) to derive a contradiction.

Proof. We suppose that $\alpha $ is algebraic and obtain a contradiction. We refer to the proof of Theorem 5.1, whose notation we freely use in the following. In particular, we use the distinguished place $v_0$ on the field $\mathbb Q(\beta )$ , the tuples $\boldsymbol a_n \in (\mathcal O)_S^{m+3}$ in (5.1) and the linear form L in (5.2), which satisfies the inequality $|L(\boldsymbol a_n)| \ll |\beta |^{-t_n}$ shown in (5.3). As before, by passing to a subsequence of $n\in \mathbb N$ , we assume that there is a set $J \subseteq \{1,\ldots ,m\}$ such that for all $n\in \mathbb N$ , we have $a_{n,j}\neq 0$ if and only if $j\in J$ . Crucially, J is non-empty in this case since it contains the indices $j_0$ and $j_1$ appearing in the non-vanishing condition. The only deviation from the basic set-up of Theorem 5.1 concerns the choice of parameter $m \in \mathbb N$ in (5.1), as we now explain.

For the index $j_0$ from the non-vanishing condition, since $d(I_{n,j},I_{n,j+1}) \ll s_n$ for all $j<j_0$ by the upper-bound part of the expanding-gaps condition, we can write $a_{n,3+j_0}$ as a polynomial in $\beta $ of height at most b and degree bounded by a constant multiple of $s_n$ . Applying Proposition 2.2, there exists a constant $c_2>0$ such that $|a_{n,3+j_0}| \gg |\beta |^{-c_2s_n}$ . By the expanding gaps condition (this time, the lower bound $d(I_{n,j},I_{n,j+1}) \gg s_n$ ), we can choose m sufficiently large that $m \geq j_0,j_1$ and for all n, the right endpoint $t_n$ of interval $I_{n,m}$ satisfies $t_n \geq (\rho +6c_2)s_n$ for all n.

Exactly as in the proof of Theorem 5.1, from the subspace theorem, we obtain a linear form

(6.4) $$ \begin{align} F(x_1,\ldots,x_{3+m}) = \alpha_1x_1+ \alpha_2x_2+\alpha_3x_3+\sum_{j \in J} \alpha_{3+j}x_{3+j} \end{align} $$

with algebraic coefficients such that $F(\boldsymbol {a}_n)=0$ for infinitely many $n\in \mathbb {N}$ , and such that $\alpha _3\neq 0$ and $\alpha _{3+j}= 0$ for all $j \in J$ .

Here is where we deviate from the proof of Theorem 5.1. Subtracting a suitable multiple of F from L, we obtain a new linear form $L'$ whose support contains $x_{3+j}$ for $j\in J$ , but not $x_3$ , and that satisfies $|L'( \boldsymbol a_n)| \ll |\beta |^{-t_n}$ for infinitely many n. Since $j_0,j_1 \in J$ (by the non-vanishing condition) the support of $L'$ contains at least two variables.

Our aim is to apply the subspace theorem to $L'$ , with distinguished index $3+j_0$ and distinguished place $v_0$ . To this end, from (5.4) and (5.5), we have

(6.5) $$ \begin{align} \prod_{(i,v)\in (\{1,2,3\} \cup J) \times S} |a_{n,i}|_v \ll |\beta|^{e_0t_n/2} \,. \end{align} $$

Recalling that $|a_{n,3+j_0}| \gg |\beta |^{-c_2s_n}$ and $|L'( \boldsymbol a_n)| \ll |\beta |^{-t_n}$ , since $t_n \geq 6c_2s_n$ , we have

(6.6) $$ \begin{align} |L'(\boldsymbol a_n)|_{v_0} \, |a_{n,3+j_0}|_{v_0}^{-1} \ll |\beta|^{-e_0t_n} |\beta|^{e_0c_2s_n} \leq |\beta|^{-5e_0t_n/6} \,. \end{align} $$

Multiplying (6.5) and (6.6), we get

(6.7) $$ \begin{align} |L'(\boldsymbol{a}_n)|_{v_0} \cdot \prod_{\substack{(i,v)\in (\{1,2,3\} \cup J) \times S\\(i,v) \neq (3+j_0,v_0)}}|a_{n,i}|_v \ll |\beta|^{-e_0t_n/3} \,. \end{align} $$

For all but finitely many n, the right-hand side of (6.7) is less than $|\beta |^{-e_0 t_n/4 }$ . However, there exists a constant $c_3>0$ such that the height of $\boldsymbol a_n$ satisfies the bound $H(\boldsymbol {a}_n) \leq |\beta |^{c_3 t_n}$ for all n. Thus, the right-hand side of (6.7) is at most $H(\boldsymbol {a}_n)^{-e_0/4c_3}$ for infinitely many n. Thus, we may apply the subspace theorem to obtain a non-zero linear form

$$ \begin{align*} G(x_1,\ldots,x_{3+m}) :=\beta_1x_1+\beta_2x_2+\sum_{j \in J} \beta_{3+j} x_{3+j}\end{align*} $$

with algebraic coefficients whose support does not include $x_3$ such that $G(\boldsymbol a_n)=0$ for infinitely many n. As shown in the proof of the claim at the end of Theorem 5.1, the conditions that $G(\boldsymbol a_n)=0$ for infinitely many n and that $x_3$ not appear in the support of G entail that G is identically zero, which yields the desired contradiction.

7.1 Balanced pairs for the k-Bonacci morphism

Write $\Sigma _{\bot }:=\Sigma \cup \{-1\}$ . For $a \in \Sigma _\bot $ , the iterated palindromic closure $\mathrm {pal}(a) \in \Sigma ^*$ is inductively defined by $\mathrm {pal}(-1):=\varepsilon $ and $\mathrm {pal}(a) := \mathrm {pal}(a-1) \,a\, \mathrm {pal}(a-1)$ for $a> -1$ . (Note that $\mathrm {pal}(a)$ denotes the iterated palindromic closure of the string $01\cdots a$ , as introduced by de Luca [Reference de Luca12].) In manipulating balanced pairs, we write expressions in the free group generated by $\Sigma $ that simplify to elements of the free monoid generated by $\Sigma $ . For example, for $x,y \in \Sigma ^*$ , we write $y^{-1}x$ when y is a prefix of x.

1. (1) For $a \in \Sigma $ , we write $[a]$ for the balanced pair $ \big[\begin{smallmatrix}a\\a\end{smallmatrix}\big] $ .

2. (2) For $a>b>c$ in $\Sigma _\bot $ , $[a,b,c]$ denotes the balanced pair $\Big[\begin{smallmatrix} a\, \mathrm {pal}(b) \mathrm {pal}(c)^{-1} \\ \mathrm {pal}(c)^{-1}\mathrm {pal}(b)\, a \end{smallmatrix}\Big]$ . This pair is irreducible since the only occurrences of a are on the top left and bottom right.

3. (3) For $a<b<c$ in $\Sigma _\bot $ , $[a,b,c]$ denotes the balanced pair $\Big[\begin{smallmatrix} \mathrm {pal}(a)^{-1}\mathrm {pal}(b)\, c \\ c\, \mathrm {pal}(b) \mathrm {pal}(a)^{-1} \end{smallmatrix}\Big]$ . This pair is irreducible since the only occurrences of c are on the top right and bottom left.

We consider the following alphabet of irreducible balanced pairs:

$$ \begin{align*} \Gamma := \{[a]:a\in \Sigma\} \cup \{[a,b,c]\in (\Sigma_\bot)^3 : a>b>c \text{ or } a<b<c\} \, .\end{align*} $$

We extend the definition of iterated palindromic closure to a map $\mathrm {pal}:\Sigma \times \Sigma \rightarrow \Gamma ^*$ , defined inductively by $\mathrm {pal}(a,a) := \varepsilon $ and

$$ \begin{align*} \mathrm{pal}(a,b) := \begin{cases} \mathrm{pal}(a-1,b)\, [a,b,-1]\, \mathrm{pal}(a-1,b) & (a>b),\\ \mathrm{pal}(a,b-1) \, [-1,a,b] \, \mathrm{pal}(a,b-1) & (a<b). \end{cases} \end{align*} $$

We use this map to characterise the lifting of $\varphi $ to a morphism $\widetilde {\varphi } :\Gamma ^+\rightarrow \Gamma ^+$ as follows:

$$ \begin{align*} \begin{array}{rcll} \widetilde{\varphi} ([a])\!\!\! & :=\!\!\!& \begin{cases} [0] \, [a+1] & \text{if } a< k-1 ,\\ [0] \, & \text{if } a=k-1, \end{cases} & \\[0.2ex] \widetilde{\varphi} ([a,b,c])\!\!\! & :=\!\!\!& \begin{cases} [0]\, [a+1,b+1,c+1] & \text {if }a< k-1, \\ [0]\, \mathrm{pal}(c+1,b+1) & \text{if }a=k-1, \end{cases} & (a>b>c)\\ \widetilde{\varphi} ([a,b,c])\!\!\! & :=\!\!\!& \begin{cases} [0]\, [a+1,b+1,c+1] & \text {if }c< k-1, \\ [0]\, \mathrm{pal}(c+1,b+1) & \text{if }c=k-1, \end{cases} & (a<b<c).\\ \end{array} \end{align*} $$

The correctness of the above characterisation can be shown by direct calculation using the facts that $\varphi (\mathrm {pal}(a)) = \mathrm {pal}(a+1)0^{-1}$ for $a\in \{0,\ldots ,k-2\}$ and $\varphi (\mathrm {pal}(-1)) = \varepsilon = \mathrm {pal}(-1)$ . For example, for $k-1>a>b>c \geq 0$ , we have

$$ \begin{align*} \varphi(a \, \mathrm{pal}(b) \, \mathrm{pal}(c)^{-1}) &= 0 \, (a+1) \, \mathrm{pal}(b+1) \, 0^{-1}\, 0 \, \mathrm{pal}(c+1)^{-1} \\ &= 0 \, (a+1) \, \mathrm{pal}(b+1)\, \mathrm{pal}(c+1)^{-1} \end{align*} $$

and

$$ \begin{align*} \varphi(\mathrm{pal}(c)^{-1} \, \mathrm{pal}(b) \, a) &= 0 \, \mathrm{pal}(c+1)^{-1} \, 0 \, 0^{-1} \, \mathrm{pal}(b+1) \, (a+1)\\ &= 0 \, \mathrm{pal}(c+1)^{-1} \mathrm{pal}(b+1) (a+1) \,. \end{align*} $$

We conclude that $\widetilde {\varphi }([a,b,c]) =[0]\, [a+1,b+1,c+1]$ . The other cases follow by similar reasoning.

The construction above shows that the balanced-pair algorithm terminates with coincidence for the k-Bonacci morphism. Using Lemma 4.4, we conclude that the k-Bonacci word is echoing. The rest of the proof is dedicated to establishing the non-vanishing condition.

The incidence graph $\mathcal G_{\widetilde {\varphi }}$ of $\widetilde {\varphi }$ has set of vertices the set $\Gamma _0 := \Gamma \setminus \{[a]: a\in \Sigma \}$ comprising the mismatch symbols $[a,b,c]$ in $\Gamma $ . There is a directed edge from $[a,b,c]$ to $[a',b',c']$ if and only if $[a',b',c']$ appears in the string $\widetilde {\varphi }([a,b,c])$ .

Proof. We specify a cycle cover $\mathcal C$ of $\mathcal G_{\widetilde {\varphi }}$ by defining a function $s:\Gamma _0\rightarrow \Gamma _0$ that determines an outgoing edge from every vertex. Specifically, for $a>b>c$ , we have

$$ \begin{align*} s([a,b,c]):= \begin{cases}[a+1,b+1,c+1], & a<k-1,\\ [-1,c+1,b+1], &a=k-1,\end{cases} \end{align*} $$

and for $a<b<c$ , we have

$$ \begin{align*} s([a,b,c]):= \begin{cases} [a+1,b+1,c+1], & c<k-1,\\ [b+1,a+1,-1], &c=k-1.\end{cases} \end{align*} $$

Informally, whenever there is a choice of an outgoing edge from a node $[a,b,c]$ in $\mathcal G_{\widetilde {\varphi }}$ , the function s selects the edge leading to the central letter of the string $\widetilde {\varphi }([a,b,c])$ .

It can be seen by direct calculation that $\mathcal C$ is a cycle cover. Indeed, $\mathcal C$ is a union of cycles of the following form, where $a>b >-1$ :

$$ \begin{align*} [a,b,-1] & \stackrel{s^{k-a}}{\longmapsto} [-1,k-a-1,k+b-a] \stackrel{s^{a-b}}{\longmapsto} [k-b-1,a-b-1,-1] \stackrel{s^{b+1 }}{\longmapsto} [-1,b,a] \\ & \stackrel{s^{k-a}}{\longmapsto} [k+b-a,k-a-1,-1] \stackrel{s^{a-b}}{\longmapsto} [-1,a-b-1,k-b-1] \stackrel{s^{b+1}}{\longmapsto} [a,b,-1] \,. \end{align*} $$

We now show that $\mathcal C$ is the unique cycle cover of $\mathcal G_{\widetilde {\varphi }}$ . To this end, recall that a $\mathcal C$ -alternating cycle is a bipartite graph $\mathcal {A}=(U,V,E)$ , where U and V are disjoint sets of vertices of $\mathcal G_{\widetilde {\varphi }}$ with $|U|=|V|\geq 2$ , $E\subseteq U\times V$ is a set of edges of $\mathcal G_{\widetilde {\varphi }}$ and each vertex of $\mathcal {A}$ is incident to two edges in E, of which one lies in $\mathcal C$ . By Berge’s theorem [Reference Lovász and Plummer23, Ch. 2], the uniqueness of $\mathcal C$ as a cycle cover is equivalent to the non-existence of a $\mathcal C$ -alternating cycle; see Figure 1 for a representation of an alternating cycle.

Figure 1

An alternating cycle for a cycle cover $\mathcal C$ . The horizontal edges lie in $\mathcal C$ and the non-horizontal edges are not in $\mathcal C$ . Removing the horizontal edges from $\mathcal C$ and replacing them by the non-horizontal edges yields a new cycle cover. However, if $\mathcal C'$ is a cycle cover other than $\mathcal C$ , then the symmetric difference of the respective sets of edges in $\mathcal C$ and $\mathcal C'$ can be partitioned into alternating paths of the above form.

The nodes in $\mathcal G_{\widetilde {\varphi }}$ that have outdegree at least two have the form $[k{\kern-1pt}-{\kern-1pt}1,a,b]$ or ${[a,b,k{\kern-1pt}-{\kern-1pt}1]}$ . Hence, for any bipartite subgraph of the form shown in Figure 1, there are two cases. The first case is that $a_0{\kern-1pt}={\kern-1pt}a_2{\kern-1pt}={\kern-1pt}\cdots {\kern-1pt}={\kern-1pt}a_{2m}{\kern-1pt}={\kern-1pt}k-1$ and $a_1{\kern-1pt}={\kern-1pt}a_3{\kern-1pt}={\kern-1pt}\cdots {\kern-1pt}={\kern-1pt} a_{2m+1}{\kern-1pt}={\kern-1pt}-1$ . Since the blue edges point to the central factor of an iterated palindromic closure, we have $c_1>c_3>\cdots > c_{2m+1}>c_1$ , which is a contradiction. The second case is that ${c_0=c_2=\cdots =c_{2m}=k-1}$ and $c_1=c_3=\cdots = c_{2m+1}=-1$ . Then, we have $a_1>a_3>\cdots > a_{2m+1}>a_1$ , which is again a contradiction. We conclude that an alternating cycle of the form shown in Figure 1 cannot exist and, hence, the cycle cover $\mathcal C$ is unique.

7.2 Strong echoing condition

Write $\Gamma _0:=\{a_0,\ldots ,a_m\}$ for the set of mismatch symbols in $\Gamma $ . For $i \in \{0,\ldots ,m\}$ , define $Q_{n,i}(x) \in \mathbb {Z}[x]$ by

$$ \begin{align*} Q_{n,i}(x) := \sum_{k=0}^{\ell-1} (\pi_1(\widetilde{\varphi}^n(a_i))_k - \pi_2(\widetilde{\varphi}^n(a_i))_k)x^{k} \,. \end{align*} $$

Using Proposition 7.1, we prove the following result.

Proof. Without loss of generality, assume that $a_0 \in \Gamma $ is the matched pair $(01,10)$ . Then, $Q_{0,0}(x)$ is the polynomial $-1+x$ . Hence, $\boldsymbol Q_0(\beta ) \neq \boldsymbol 0$ for $\beta \neq 1$ .

We next describe a recurrence $\boldsymbol Q_{n+1} = M_n \, \boldsymbol Q_n$ , where $M_n \in \mathbb Z[x]^{(m+1)\times (m+1)}$ . The matrix entries $(M_n)_{i,j}$ are defined as follows. Suppose $\widetilde {\varphi }(a_i) = a_{j_1} \cdots a_{j_s}$ . Then, $\widetilde {\varphi }^{n+1}(a_i) = \widetilde {\varphi }^n(a_{j_1}) \cdots \widetilde {\varphi }^n(a_{j_s})$ . This equation allows us to write the polynomial $Q_{n+1,i}(x)$ as a linear combination of the polynomials $Q_{n,0}(x),\ldots ,Q_{n,m}(x)$ . Specifically, we have

$$ \begin{align*} (M_n)_{i,j} := \sum_{k:j_k=j} x^{f(k)} \quad\text{where } f(k) = |\pi_1( \widetilde{\varphi}^n(a_{j_1}) \cdots \widetilde{\varphi}^n(a_{j_k-1}))| \,. \end{align*} $$

Note that $(M_n)_{i,j}\neq 0$ only if there is an edge from $a_i$ to $a_j$ in the incidence graph $\mathcal G_{\widetilde {\varphi }}$ . Thus, we have $\det (M_n) = \prod _{i=0}^m (M_n)_{i,s(i)}$ for the map $s : \{0,\ldots ,m\}\rightarrow \{0,\ldots ,m\}$ that specifies the unique cycle cover of $\mathcal G_{\widetilde {\varphi }}$ , as defined in the proof of Proposition 7.1. By definition of s, the symbol $a_{s(i)}$ has a single occurrence in the string $\widetilde {\varphi }(a_i)$ , namely as the central factor in an iterated palindromic closure. It follows that $M_{i,s(i)}$ is a monomial for all $i\in \{1,\ldots ,m\}$ and, hence, $\det (M_n)$ is a monomial. Thus, $M_n(\beta ^{-1})$ is non-singular for all $\beta \neq 0$ .

From recurrence $\boldsymbol Q_{n+1} = M_n \, \boldsymbol Q_n$ and the facts that $\boldsymbol Q_0(\beta ) \neq \boldsymbol 0$ for $\beta \neq 1$ and $\det (M_n)(\beta )\neq 0$ for $\beta \neq 0$ , we have by induction on $n\in \mathbb N$ that $\boldsymbol Q_n(\beta ^{-1})\neq \boldsymbol 0$ for all $n\in \mathbb N$ and $\beta \in \mathbb C\setminus \{0,1\}$ .

We can now prove the main result of this section. The proof develops the construction in §§4.2 and 6.2 showing that the Tribonacci word is strongly echoing.

Proof of Theorem 7.1

By inspection, the lifting $\widetilde {\varphi }$ of the k-Bonacci morphism $\varphi $ to the set $\Gamma $ of balanced pairs satisfies the coincidence condition. Furthermore, since $\varphi $ is primitive, there exists $c_0>0$ such that for all $\gamma ,\gamma ' \in \Gamma $ and all $n\in \mathbb N$ , we have

(7.1) $$ \begin{align} |\pi_1(\widetilde{\varphi}^n(\gamma))| \leq c_0\, |\pi_1(\widetilde{\varphi}^n(\gamma'))| \,. \end{align} $$

Let $\boldsymbol u \in \Sigma ^\omega $ be the k-Bonacci word and let $\boldsymbol w \in \Gamma ^\omega $ be the image of $\boldsymbol u$ under the coding $\iota :\Sigma \rightarrow \Gamma $ given by

$$ \begin{align*} \iota(0) := \left[\begin{array}{@{}c@{\;}c@{}} 0&1\\ 1&0\end{array}\right] \quad \iota(1) := \left[\begin{array}{@{}c@{\;}c@{}} 0&2\\ 2&0 \end{array}\right] \quad \cdots\quad \iota(k-2) := \left[\begin{array}{@{}c@{\;}c@{}} 0&k-1\\ k-1&0 \end{array}\right]\quad \iota(k-1) := \left[\begin{array}{@{}c@{\;}c@{}} 0\\0 \end{array}\right] \, . \end{align*} $$

Since $\pi _1\circ \iota = \varphi $ , we have $\pi _1(\boldsymbol w)=\boldsymbol u$ . Moreover, since $\sigma \circ \pi _1\circ \iota $ and $\pi _2 \cdot \iota $ coincide on any word on $\Sigma ^\omega $ , we have $\pi _2(\boldsymbol w) = \sigma (\boldsymbol u)$ .

Given $n_0\in \mathbb N$ , write

(7.2) $$ \begin{align} \widetilde{\varphi}^{n_0}(\boldsymbol w) = w_0 a_{i_1} {w}_{1} a_{i_2} w_{2} a_{i_3} \cdots\,, \end{align} $$

where $ {w}_0, {w}_1,\ldots \in \Gamma ^*$ are sequences of coincidence symbols and $a_{i_1},a_{i_2},\cdots \in \Gamma $ are mismatch symbols. The word $\widetilde {\varphi }^{n_0}(\boldsymbol w)$ is uniformly recurrent, being the morphic image of the uniformly recurrent word $\boldsymbol u$ . Hence, there is a uniform upper bound on the length of the factors $ w_i$ , depending only on $n_0$ . Let $\varepsilon>0$ be as in the definition of echoing sequence (Definition 6.1). Since $\widetilde {\varphi }$ satisfies the coincidence condition, for sufficiently large $n_0\in \mathbb {N}$ , the upper asymptotic density of mismatch symbols in $\widetilde {\varphi }^{n_0}(\boldsymbol {w})$ is at most $\varepsilon /c_0$ .

The data to show the strong echoing property are as follows. For all $n\in \mathbb {N}$ , consider the image of (7.2) under the n-fold application of $\widetilde {\varphi }$ :

(7.3) $$ \begin{align} \widetilde{\varphi}^{n+n_0}(\boldsymbol{w}) = \widetilde{\varphi}^{n}(w_0) \underbrace{\widetilde{\varphi}^{n}(a_{i_1})}_{I_{n,1}} \widetilde{\varphi}^{n}(w_1) \underbrace{\widetilde{\varphi}^{n}(a_{i_2})}_{I_{n,2}} \widetilde{\varphi}^{n}(w_2) \underbrace{\widetilde{\varphi}^{n}(a_{i_3})}_{I_{n,3}} \cdots \,. \end{align} $$

We define $r_n:=0$ and $s_n:=|\varphi ^{n+n_0}(0)|$ . Then, $\pi _1(\widetilde {\varphi }^{n+n_0}(\boldsymbol w)) =\boldsymbol u$ and $\pi _2(\widetilde {\varphi }^{n+n_0}(\boldsymbol w)) =\sigma ^{s_n}(\boldsymbol u)$ . For all $j\in \{1,2,\ldots \}$ , we define $I_{n,j} \subseteq \mathbb {N}$ to be the smallest interval of positions in $\boldsymbol u=\pi _1(\widetilde {\varphi }^{n+n_0}(\boldsymbol w))$ that contains all mismatch symbols occurring in the factor $\pi _1(\widetilde {\varphi }^{n}(a_{i_j}))$ , as shown in (7.3). By construction, this choice satisfies the covering property. To establish the density property, we note that since the mismatch symbols in $\widetilde {\varphi }^{n_0}(\boldsymbol w)$ have upper asymptotic density at most $\varepsilon /c_0$ , by (7.1), we have $\mathrm {den}(\bigcup _{j=1}^m I_{n,j})\leq \varepsilon $ for all sufficiently large m. Finally, the uniform upper bound on the length of the factors $w_i$ and the bound (7.1) imply that $d(I_{n,j},I_{n,j+1}) \asymp s_n$ for all $j\geq 1$ . (Note that this growth bound holds even when the word $w_j$ is empty since $\widetilde {\varphi }(a_{i_{j+1}})$ starts with a coincidence.) We have thus established the expanding-gaps property.

It remains to consider the non-vanishing property. To this end, for all $n\in \mathbb N$ and $j\in \{1,2,\ldots \}$ , define the polynomial $P_{n,j}(x) \in \mathbb {Z}[x]$ by

(7.4) $$ \begin{align} P_{n,j}(x) := \sum_{i \in I_{n,j}} (u_{i+s_n}-u_{i+r_n})x^{i} \,. \end{align} $$

Given $\beta \in \mathbb C$ with $|\beta |>1$ , we must show that there exist distinct indices $j_0,j_1$ such that $P_{n,j_0}(\beta ^{-1})\neq 0$ and $P_{n,j_1}(\beta ^{-1})\neq 0$ for infinitely many $n\in \mathbb N$ . By Proposition 7.2, for all $n\in \mathbb N$ , there exists $i\in \{0,\ldots ,m\}$ such that $Q_{n,i}(\beta ^{-1})\neq 0$ . Hence, there exists $i^*$ such that $Q_{n,i^*}(\beta ^{-1})\neq 0$ for infinitely many $n\in \mathbb N$ . There are arbitrarily large choices of $n_0$ such that $i^*$ occurs infinitely often in the list $i_1,i_2,i_3,\ldots $ in (7.2). For such an $n_0$ , there exist $j_0<j_1$ such that $i_{j_0}=i_{j_1}=i^*$ . Then, $Q_{n,i_{j_0}}(\beta ^{-1})\neq 0$ and $Q_{n,i_{j_1}}(\beta ^{-1})\neq 0$ for infinitely many n. However, for all $j \in \{1,2,\ldots \}$ , $P_{n,j}$ is the product of $Q_{n,i_j}$ and a monomial in x. Hence, $P_{n,{j_0}}(\beta ^{-1})\neq 0$ and $P_{n,{j_1}}(\beta ^{-1})\neq 0$ for infinitely many n.