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The continued fraction expansion of \(\alpha\) with \(\mu(\alpha)=3\). (English) Zbl 0967.11024

For any real \(\alpha\), define \(\mu(\alpha)\) by \(1/\mu(\alpha)=\limsup_{q\to\infty}q\|q\alpha\|\), where \(q\) is an integer and \(\|\cdot\|\) is the distance from the nearest integer. The set \(\{\mu(\alpha)\}\) is called the Lagrange spectrum. In this paper the author carries out a study of the continued fraction expansions of \(\alpha\) with \(\mu(\alpha)\leq 3\).
Let \(W(a,b)\) be the set of finite words in two symbols \(a\) and \(b\). For \(W=W_0 W_1\dots\) (\(W_i\in\{a,b\}\)), define \([W]\in\mathbb R\) by the continued fraction expansion \([W]=[0; W_0, W_1, W_2, \dots]\). For \(0\leq x\leq 1\), define a one-sided infinite word \(H(x)=G(x,1)G(x,2)\dots\), a two sided infinite word \(G(x)=\dots G(x,-1)G(x,0)G(x,1)G(x,2)\dots\), where \(G(x,n)=\lfloor{nx\rfloor}-\lfloor{(n-1)x\rfloor}\). Let \(F_N\) be the Farey sequence for a natural number \(N\). For a rational \(x=n/m\neq 0\) with \((n,m)=1\), define an infinite word \(\underline{G(x)}\) by \[ \underline{G(x)}=\dots G(x,-1)G(x,0)G(u,1)\dots G(u,k)G(x,1)G(x,2)\dots, \] where \(u=\max\{y\in F_m\mid y<x\}\) and \(k\) is the denominator of \(u\) (set \(k=1\) if \(u=0\)). For a rational \(x=n/m\neq 1\) with \((n,m)=1\), define \[ \overline{G(x)}=\dots G(x,-1)G(x,0)G(u,1)\dots G(u,k)G(x,1)G(x,2)\dots, \] where \(u=\min\{y\in F_m\mid x<y\}\) and \(k\) is the denominator of \(u\) (set \(k=1\) if \(u=1\)).
Define a substitution \(\phi: W(0,1)\to W(1,2)\) by \(\phi:0\to 11, 1\to 22\). For any natural number \(N\) and an infinite word \(S\) in \(W(a,b)\), define \[ D_S(N)=\{p\in W(a,b)\mid \text{\(p\) is a subword of \(S\) and \(|p|=N\)}\} \] and \[ D_S'(N)=\{p\in W(a,b)\mid \text{\(p\) occurs infinitely many times in \(S\) and \(|p|=N\)}\} \] Then the author introduces the new sequence \(S\in W(0,1)\) called super Bernoulli sequence related to \((x,y)\) if it satisfies one of the following conditions from (1) to (4) for all natural numbers \(N\).
(1) \(D_S'(N)=\bigcup_{z\in[x,y]} D_{G(z)}(N)\),
(2) \(x\in\mathbb Q\) and \(D_S'(N)=\bigcup_{z\in[x,y]} D_{G(z)}(N)\cup D_{\underline{G(x)}}(N)\),
(3) \(y\in\mathbb Q\) and \(D_S'(N)=\bigcup_{z\in[x,y]} D_{G(z)}(N)\cup D_{\overline{G(y)}}(N)\),
(4) \(x, y\in\mathbb Q\) and \(D_S'(N)=\bigcup_{z\in[x,y]} D_{G(z)}(N)\cup D_{\underline{G(x)}}(N)\cup D_{\overline{G(y)}}(N)\).
The main result is as follows. Let \(\alpha=[a_0; a_1, \dots]\) be an irrational number with \(\mu(\alpha)\leq 3\). Then there exists a non-negative integer \(n\) such that \(a_m\in\{1,2\}\) for all \(m\geq n\), and there exists a one-sided word \(S\in W(0,1)\) which is a super Bernoulli sequence related to \((x,y)\) for some \(x\), \(y\) with \(0\leq x\leq y\leq 1\) such that \(D_A'(N)=D_{\phi(S)}(N)\) for all \(N\in\mathbb N\), where \(A=a_n a_{n+1}a_{n+2}\dots\). Conversely, let \(S\) be any super Bernoulli sequence related to \((x,y)\) and let \(A\in W(1,2)\) be a one-sided infinite word such that \(D_A'(N)=D_{\phi(S)}(N)\) for all \(N\). Then \(\mu([A])\leq 3\), and strict inequality holds if and only if \(x=y\) is rational and \(S\) is a super Bernoulli sequence satisfying the condition (1) above.
The author also proves the existence and some properties of super Bernoulli sequences.

MSC:

11J06 Markov and Lagrange spectra and generalizations
11J70 Continued fractions and generalizations
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