UDC 511.7

MATHEMATICS

A. A. BERSHTEIN

ON NECESSARY AND SUFFICIENT CONDITIONS FOR POINTS OF THE MARKOV SPECTRUM TO BELONG TO THE LAGRANGE SPECTRUM

(Presented by Academician Yu. V. Linnik on 2 IX 1969)

The present paper is a development of the ideas and methods of the work of G. A. Freiman (¹).

Notation: \(x\) is a sequence of natural numbers \(a_0, \ldots, a_i, \ldots\);

\[ [q_1; q_2, \ldots] = q_1 + \frac{1}{q_2+\ldots}; \qquad \lambda_i(\mathscr L)=a_i+[0; a_{i+1}, \ldots]+[0; a_{i-1}, \ldots, a_0]; \]

\[ \lambda(\mathscr L)=\varlimsup_{i\to\infty}\lambda_i(\mathscr L); \]

\(\{\lambda(\mathscr L)\}\) is the Lagrange spectrum; \(M\) is a sequence of natural numbers infinite in both directions
\(\ldots, a_{-1}, a_0, a_1, \ldots\);
\[ \lambda_i(M)=a_i+\delta_i+\gamma_i, \]
where
\[ \delta_i=[0; a_{i+1}, a_{i+2}, \ldots]; \qquad \gamma_i=[0; a_{i-1}, a_{i-2}, \ldots]; \qquad \lambda(M)=\sup_i \lambda_i(M); \]
\(\{\lambda(M)\}\) is the Markov spectrum;
\[ \alpha=4\sqrt{30}/7, \qquad \beta=\sqrt{689}/8. \]

A sequence \(M_1, M_2, \ldots, M_j, \ldots\) will be called convergent to the sequence \(M\) if, for any integer \(i\), we have

\[ \lim_{j\to\infty} a_i^{(j)} = a_i, \qquad \lim_{j\to\infty} \lambda(M_j)=\lambda(M). \]

A sequence \(\{M_j\}\), \(j=1,2,\ldots\), will be called stabilizing from the right (from the left) if there exist a natural number \(j_0\) and an integer \(i_0\) such that, for \(i>i_0\) and \(j>j_0\) (respectively \(i<i_0\)),

\[ a_i^{(j)}=a_i. \tag{1} \]

Theorem 1. Let \(\lambda(M)\in(\alpha,\beta)\). If there exists a sequence \(\{M_j\}\) converging to \(M\) and not stabilizing either from the right or from the left, then \(\lambda(M)\in\{\lambda(\mathscr L)\}\).

Proof. Without loss of generality, we assume that \(\lambda(M)=\lambda_0(M)\). Let, for the given \(j\), condition (1) be satisfied for
\[ -i_0 \le i \le i_1; \qquad i_0,i_1>0, \]
\[ a_{-i_0-1}^{(j)}\ne a_{-i_0-1}, \qquad a_{i_1+1}^{(j)}\ne a_{i_1+1}. \]
Define the sequence
\[ \bar M_j=\{\ldots,\bar a_{-1},\bar a_0,\bar a_1,\ldots\} \]
by the equalities:
\[ \bar a_i=a_i,\quad -i_0\le i\le i_1; \qquad \bar a_{i_1+1}=\bar a_{i_1+2}=2; \qquad \bar a_{i_1+3}=1; \qquad \bar a_{i_1+4}=\bar a_{i_1+5}=\bar a_{i_1+6}=2; \]
\[ \bar a_i=\bar a_{i-4},\quad i\ge i_1+6; \qquad \bar a_{-i_0-1}=\bar a_{-i_0-2}=2; \qquad \bar a_{-i_0-3}=1; \]
\[ \bar a_{-i_0-4}=\bar a_{-i_0-5}=\bar a_{-i_0-6}=2; \qquad \bar a_i=\bar a_{i+4},\quad i\le -i_0-6, \]
i.e.
\[ \bar M_j=\{\ldots,2,2,2,1,2,2,a_{-i_0},\ldots,a_{i_1},2,2,1,2,2,2,\ldots\}. \]

For this sequence, when \(i>i_1\), \(i<-i_0\), we have

\[ \lambda_i(\bar M_j)\le \alpha, \tag{2} \]

since for the combinations
\[ \left\{\begin{matrix}222\\ i\end{matrix}\right\},\quad \left\{\begin{matrix}122\\ i\end{matrix}\right\} \quad\text{and}\quad \left\{\begin{matrix}212\\ i\end{matrix}\right\} \]
we obtain the estimate \(\lambda_i\le\alpha\), if one takes into account that the presence in \(M\) of the combinations \(\{3\}\) and \(\{1212\}\) gives \(\lambda(M)>\beta\).

Let now \(-i_0 \leq i \leq i_1\). We shall show that

\[ \delta_i(\overline{M}_j) \leq \max(\delta_i(M),\ \delta_i(M_j)), \tag{3} \]

\[ \gamma_i(\overline{M}_j) \leq \max(\gamma_i(M),\ \gamma_i(M_j)). \tag{4} \]

We shall prove inequality (3), since (4) is proved analogously. Let \(i\) have the same parity as \(i_1-1\). Suppose, for example, that \(a_{i_1+1}=2\), \(a^{(j)}_{i_1+1}=1\). Then

\[ [0;\ a_i,\ \overline{a}_{i+1},\ldots,a_{i_1},\ 2,\ldots] < [0;\ a_i,\ldots,a_{i_1},\ 1,\ldots] \]

for any \(a_i,\ i<i_1\). Therefore

\[ \delta_i(\overline{M}_j) \leq \delta_i(M_j). \tag{5} \]

Let now \(i\) have the same parity as \(i_1\). If \(a_{i_1+2}=1\), then

\[ [0;\ a_i,\ldots,a_{i_1},\ 2,\ 2,\ldots] < [0;\ a_i,\ldots,a_{i_1},\ 2,\ 1,\ldots]. \tag{6} \]

If \(a_{i_1+2}=2,\ a_{i_1+3}=2\), then

\[ [0;\ a_i,\ldots,a_{i_1},\ 2,\ 2,\ 1,\ldots] < [0;\ a_i,\ldots,a_{i_1},\ 2,\ 2,\ 2,\ldots]. \tag{7} \]

If \(a_{i_1+2}=2,\ a_{i_1+3}=1,\ a_{i_1+4}=1\), then

\[ [0;\ a_i,\ldots,a_{i_1},\ 2,\ 2,\ 1,\ 2,\ldots] < [0;\ a_i,\ldots,a_{i_1},\ 2,\ 2,\ 1,\ 1,\ldots]. \tag{8} \]

If \(a_{i_1+2}=2,\ a_{i_1+3}=1,\ a_{i_1+4}=a_{i_1+5}=2\), then \(a_{i_1+5}=2\), since otherwise the combination \(\left\{\begin{smallmatrix}2121\\ i\end{smallmatrix}\right\}\) would occur in \(M\). If \(a_{i_1+2}=2,\ a_{i_1+3}=1,\ a_{i_1+4}=a_{i_1+5}=2\) and \(a_{i_1+6}=1\), then

\[ [0;\ a_i,\ldots,a_{i_1},\ 2,\ 2,\ 1,\ 2,\ 2,\ 2,\ldots] < [0;\ a_i,\ldots,a_{i_1},\ 2,\ 2,\ 1,\ 2,\ 2,\ 1,\ldots]. \tag{9} \]

Each of the inequalities (6), (7), (8), (9) entails the validity of the inequality

\[ \delta_i(\overline{M}_j)<\delta_i(M). \]

Since the length of the period in \(\overline{M}_j\) is equal to 4, inequality (3) is proved completely.

From (1)—(4) it follows that

\[ \lim_{j\to\infty}\lambda(\overline{M}_j)=\lambda(M). \tag{10} \]

Let

\[ T_j,\ R_j \geq \max(2i_0,\ 2i_1), \tag{11} \]

with \(a^{(j)}_{R_j}=1\) and \(\overline{a}^{(j)}_{-T_j}=\overline{a}^{(j)}_{-T_j+1}=\overline{a}^{(j)}_{-T_j+2}=2\).

For the sequence

\[ \mathcal{L}=\{a^{(1)}_{-T_1},\ldots,a^{(1)}_{R_1},\ a^{(2)}_{-T_2},\ldots,a^{(2)}_{R_2},\ldots\} \]

in view of (10) and (11),

\[ \lambda(\mathcal{L})=\lambda(M). \]

A direct consequence of Theorem 1 is

Theorem 2. Let \(\lambda(M)\in(\alpha,\beta)\), and suppose there exists an infinite set of sequences \(M'\) for which \(\lambda(M')=\lambda(M)\). Then \(\lambda(M)\in\{\lambda(\mathcal{L})\}\).

We say that for some \(\varepsilon>0\) the sequence \(M\) has the \(\varepsilon\)-property on the right (on the left) if there exists an integer \(i_0\) such that for any sequence \(M'\ne M\) for which \(a_i=a'_i\) for \(i<i_0\) (respectively \(i>i_0\)), we have \(\lambda(M')>\lambda(M)+\varepsilon\). If \(\lim\limits_{i\to+\infty}\lambda_i(M)<\lambda(M)\), then the sequence is called indefinite.

Theorem 3. Let \(\lambda(M_1)\in(\alpha,\beta)\). In order that \(\lambda(M_1)\in\overline{\{\lambda(\mathcal L)\}}\), the following conditions are necessary and sufficient:

1. There exists only a finite number of sequences \(\overline{M_s}\), \(1\le s\le k\), such that
   \[ \lambda(\overline{M_1})=\lambda(\overline{M_2})=\ldots=\lambda(\overline{M_k}). \]

2. The sequences \(M_s\) are nonperiodic.

3. The sequences \(M_s\) are periodic on the right*, i.e., there exist an integer \(i_0\) and a natural number \(p\) such that, for \(i>i_0\), \(a_i=a_{i+p}\).

4. Each of the sequences \(M_s\), periodic in both directions, with period equal to the period of the sequence \(M_s\), has the \(\varepsilon\)-property on the right, where
   \[ \varepsilon=\lambda(\overline{M_s})-\lim_{i\to\infty}\lambda_i(M_s). \]

Proof. The necessity of condition 1 follows from Theorem 2. Since the sequence \(M_1\) is nonperiodic (see (2)), there exist an integer \(i_1\) and \(\varepsilon_1>0\) such that \(\lambda_i(M_1)<\lambda(M_1)-\varepsilon_1\) for \(i>i_1\). One can choose an \(s=s(\varepsilon_1)\) such that if in \(M_1\) and \(M'\) we have \(a_i=a_i'\) for \(i\le j+s\) or \(i\ge j-s\), then
\[ \lambda_j(M')<\lambda_j(M)+\frac{\varepsilon_1}{2}. \tag{12} \]

There will be natural numbers \(b_1,b_2,\ldots,b_{2s}\) and sequences \(i_1,i_2,\ldots,i_r,\ldots\) such that \(a_{i_r+t}=b_{t+1}\), \(t=0,1,\ldots,2s-1\), \(r=1,2,\ldots\). Form the sequences
\[ M^{(r)}=\{a_i^{(r)}=a_i,\ i\le i_r,\ a_i^{(r)}=a_{i+i_1-i_r},\ i>i_r\}. \tag{13} \]

For them, in view of (12) and (13),
\[ \lambda_i(M^{(r)})<\lambda_i(M_1)+\frac{\varepsilon_1}{2},\quad i\le i_r+s. \]

In view of (12),
\[ \lambda_i(M^{(r)})<\lambda(M_1),\quad i>i_r+s. \]

Analogous arguments can also be carried out for small \(i\). If we assume that \(M_1\) is nonperiodic both on the right and on the left, then it would now follow from Theorem 1 that \(\lambda(M_1)\in\overline{\{\lambda(\mathcal L)\}}\).

Finally, condition 4 is also satisfied, since otherwise it would follow from Theorem 1 that \(\lambda(M_1)\in\overline{\{\lambda(\mathcal L)\}}\).

Conditions 1–4 are also sufficient in order that \(\lambda(M_1)\in\overline{\{\lambda(\mathcal L)\}}\). Indeed, if \(\lambda(M_1)=\lambda(\mathcal L)\) for some \(\mathcal L\), then in \(\mathcal L\) there would occur arbitrarily long segments of the sequences \(M^{(r)}\), and this contradicts conditions 1–4.

All the theorems of the present paper are formulated only for \(\lambda(M)\in(\alpha,\beta)\). The set for which Theorems 1–3 are valid can be substantially enlarged.

Central Economics and Mathematics Institute
Academy of Sciences of the USSR
Moscow

Received
14 VI 1969

CITED LITERATURE

1. G. A. Freiman, Mathematical Notes, 3, No. 2, 195 (1968).
2. P. G. Kogonia, Proceedings of the Tbilisi Mathematical Institute, 29, 15 (1963).

* The sequence \(M\) can always be replaced by \(M'\), for which \(b_i=a_{-i}\), and we shall not distinguish between two such sequences.