Mathematics

P. G. Kogoniya

On Limit Points of the Set of Markov Numbers

(Presented by Academician A. N. Kolmogorov on 10 VII 1957)

I. Let (\alpha) be an arbitrary irrational number of the interval ((0, 1)),

[
\alpha = [0; \ a_1, a_2, \ldots, a_k, \ldots]
\tag{1}
]

its expansion into an arithmetic continued fraction.
Let (M(N)) be the set of all numbers (\alpha) for which

[
\varlimsup_{k \to \infty} a_k = N \qquad (N = 1, 2, \ldots);
\tag{2}
]

then

[
M = \sum_{N=1}^{\infty} M(N)
\tag{3}
]

is the set of all irrational numbers of the interval ((0, 1)) with a bounded sequence of partial quotients (a_k).

Denote by (L(\alpha)) the lower bound of the set of all real numbers (c > 0) for which the inequality

[
\left| \alpha - \frac{p}{q} \right| < \frac{c}{q^2}
\tag{4}
]

has infinitely many solutions in integers (q > 0, p).
As is known ((¹), p. 29),

[
L(\alpha) =
\frac{1}{
\varlimsup\limits_{k \to \infty}
\bigl([a_k; \ a_{k+1}, \ldots] + [0; \ a_{k-1}, a_{k-2}, \ldots, a_1]\bigr)
}.
\tag{5}
]

For brevity, let us introduce the notation:

[
\lambda_k(\alpha) =
[a_k; \ a_{k+1}, \ldots] + [0; \ a_{k-1}, a_{k-2}, \ldots, a_1],
\tag{6}
]

[
\lambda(\alpha) = \varlimsup_{k \to \infty} \lambda_k(\alpha).
\tag{7}
]

From (5), (6), and (7) it follows that

[
L(\alpha) = \frac{1}{\lambda(\alpha)}.
\tag{8}
]

Let (M_L(N)) be the set of values of (L(\alpha)) as (\alpha) ranges over the whole set (M(N)); then

[
M_L = \sum_{N=1}^{\infty} M_L(N)
\tag{9}
]

is the set of all values of the function (L(\alpha)). The values of the function (L(\alpha)) will be called by us Markov numbers, so that, in this terminology, (M_L) is the set of all Markov numbers.

Several facts are known about the set of Markov numbers (see (²), pp. 121–132). The present paper is devoted to a further investigation of the structure of this set, more precisely, of its subsets (M_L(N)).

Lemma. If ({k_n}) is a sequence of indices of extreme positions of the number (N) in the expansion (\alpha), (\alpha \in M(N)), then for (N=2,3)

[
\lambda(\alpha)=\overline{\lim_{k\to\infty}}\lambda_k(\alpha)
=\overline{\lim_{n\to\infty}}\lambda_{k_n}(\alpha).
\tag{10}
]

Theorem 1. The minimal point of the set (M_L(N)) ((N \geqslant 2)) is an accumulation point of this set.

Proof. As is known ((¹), p. 33), the set (M_L(N)) is contained in the segment
[
[(N^2+4N)^{-1/2},\ (N^2+4)^{-1/2}],
]
and the endpoints of this segment belong to the set (M_L(N)). Consequently, it is required to prove that the number ((N^2+4N)^{-1/2}) is an accumulation point of (M_L(N)).

1) (N \geqslant 4). Consider the expression
[
V_{l,m}(x,y)=[N;\ (1,N)_l,N,x]+[0;\ (1,N)_m,N,y], \quad
1\leqslant x,y\leqslant N+1.
\tag{11}
]

From (11) it follows that, uniformly with respect to (x,y) in the indicated domain, the equality holds

[
\lim_{l,m\to\infty} V_{l,m}(x,y)
= N+2[0;\ (1,N)_\infty]=(N^2+4N)^{-1/2},
\tag{12}
]

where

[
V_{l,m}(1,1)\leqslant V_{l,m}(x,y)\leqslant
V_{l,m}(N+1,N+1)\leqslant (N^2+4N)^{-1/2},
\tag{13}
]

as follows directly from the properties of continued fractions.

Let (\varepsilon>0) be any prescribed arbitrarily small number; then, on the basis of (12) and (13), there exists (N_1=N_1(\varepsilon)) such that for (l,m>N_1), for any pair (x,y) from the indicated domain, the inequality holds

[
(N^2+4N)^{-1/2}-\varepsilon
< V_{l,m}(x,y) < (N^2+4N)^{-1/2}.
\tag{14}
]

Next, let (\overline{\mathfrak M}(N-2)) be the set of all numbers

[
\beta=[0;\ b_1,b_2,\ldots,b_k,\ldots],
]

defined by the condition

[
1\leqslant b_k\leqslant N-2 \quad (k\geqslant 1).
\tag{15}
]

(\overline{\mathfrak M}(N-2)) has the cardinality of the continuum.

Let (l_0,m_0>N_1) be a fixed pair of natural numbers; then, by virtue of (14),

[
(N^2+4N)^{-1/2}-\varepsilon
< V_{l_0,m_0}(x,y) < (N^2+4N)^{-1/2}.
\tag{16}
]

Consider the finite ordered system of natural numbers

[
{N,\ (N,1){l_0},\ N_1(1,N),\ N}=T
\tag{17}
]

and some infinite increasing sequence ({t_n}) of natural numbers, and define the number (\alpha) by the expansion

[
\alpha=[0;\ b_1,b_2,\ldots,b_{t_1}, b_{t_1}, b_{t_1-1},\ldots,b_1,T,b_1;b_2,\ldots,b_{t_2}, b_{t_2}, b_{t_2-1},\ldots,b_1,T,\ldots]
]
[
=[0;\ a_1,a_2,\ldots,a_k,\ldots].
\tag{18}
]

(\alpha\in M(N)), and the set of all such (\alpha) has the cardinality of the continuum.

Further, by means of an argument based on the same idea as in (2), it is established that all these values (L(\alpha)) are distinct and lie in the segment
[
\left[\frac{1}{\sqrt{N^2+4N}},\ \frac{1}{\sqrt{N^2+4N-\varepsilon}}\right],
]
whence the validity of the theorem follows for (N\geqslant 4).

2) In the case (N=2) or (N=3) the validity of the theorem is established by analogous arguments; only the form of the constructed numbers (\alpha) is changed in the corresponding way, and the lemma is used essentially.

Theorem 2. The maximal point ((N^2+4)^{-1/2}) of the set (M_L(N)) ((N>1)) is its isolated point.

II. Theorem 3. The maximal point of condensation of the set (M_L(3)) is equal to
[
\frac{22}{65+9\sqrt{3}}.
]

Proof. It is known ((1), p. 33) that to the right of the number
[
\frac{22}{65+9\sqrt{3}}
]
there is the unique number
[
\frac{1}{\sqrt{13}}
]
of the set (M_L(3)). Therefore it is enough to establish that in an arbitrarily small neighborhood of the number
[
\frac{22}{65+9\sqrt{3}}
]
there is a continuum subset of the set (M_L(3)).

Let ({m_k}) and ({n_k}) be two arbitrary sequences of natural numbers tending to (\infty) as (k\to\infty); then, for the number (\alpha) defined by the expansion
[
\alpha=[0;(1,2){m_1},3,3,(2,1),(1,2){m_2},3,3,(2,1),(1,2){m_3},3,3,(2,1),\ldots]
]
[
=[0;a_1,a_2,\ldots],\qquad \alpha\in M(3),
\tag{19}
]
on the basis of (6), (7), and (8), the equalities
[
\lambda(\alpha)=[3;(2,1)\infty]+([3;(2,1)\infty])^{-1}
=\frac{65+9\sqrt{3}}{22}=A,
\tag{20}
]
[
L(\alpha)=\frac{1}{\lambda(\alpha)}=\frac{1}{A}=\frac{22}{65+9\sqrt{3}}.
\tag{21}
]
hold.

Next, let
[
U_{l,m}(x,y)=[3;(2,1)l,x]+[0;3,(2,1)_m,y],\qquad 1\leq x,y\leq 3.
\tag{22}
]
From (20) and (22) it follows that in the rectangle ([1,3;\,1,3]) the equality
[
\lim(x,y)=A} U_{l,m
\tag{23}
]
holds uniformly with respect to (x,y).

Let (\varepsilon) be an arbitrarily small positive number; then, by virtue of (23), for (l,m>N=N(\varepsilon)) the inequalities
[
A-\varepsilon<U_{l,m}(x,y)<A+\varepsilon,\qquad 1\leq x,y\leq 3.
\tag{24}
]
will hold.

If (l_0>N) is any fixed natural number, then for all sufficiently large (m_0) ((m_0>N)) the sum
[
\left[0;(2,1){l_0},\frac{1}{\beta}\right]
+
\left[0;3,(2,1)\right]},\frac{1}{\beta
=
U_{l_0,m_0}\left(\frac{1}{\beta},\frac{1}{\beta}\right)-3
\tag{25}
]
is an increasing function of (\beta), by virtue of the corresponding properties of continued fractions.

Supposing (l_0) and (m_0) to have been chosen in the manner indicated above, consider the system
[
{(1,2){l_0},\,3,\,3(2,1)}=T
\tag{26}
]

and the set (\overline{\mathfrak M}(2)) of all numbers (\beta) defined by the conditions

[
\beta=[0;\, b_1,b_2,\ldots,b_k,\ldots], \qquad 1 \leq b_k \leq 2;
\tag{27}
]

(\overline{\mathfrak M}(2)) is a set of the cardinality of the continuum.

Taking an arbitrary sequence ({t_n}) of natural numbers tending to infinity as (n\to\infty), to each (\beta), (\beta\in\overline{\mathfrak M}(2)), we assign a number (\alpha), (\alpha\in M(3)), defined by the expansion

[
\begin{aligned}
\alpha={}&[0;\, b_1,b_2,\ldots,b_{t_1}, b_{t_1}, b_{t_1-1},\ldots,b_1,T,\,
b_1,b_2,\ldots,b_{t_2}, b_{t_2}, b_{t_2-1},\ldots \
&\ldots,b_1,T,b_1,b_2,\ldots]=[0;\, a_1,a_2,\ldots];
\end{aligned}
\tag{28}
]

the set (E={\alpha}) of all such (\alpha) has the cardinality of the continuum.

By arguments analogous to those used in the preceding paper, it is established that (L(\alpha)) for (\alpha\in E) is represented as follows:

[
L(\alpha)=
\begin{cases}
\psi_1(\beta), & \text{for } \beta\in\overline{\mathfrak M}_1(2),\
\psi_2(\beta), & \text{for } \beta\in\overline{\mathfrak M}_2(2),
\end{cases}
\tag{29}
]

where

[
\overline{\mathfrak M}_1(2)+\overline{\mathfrak M}_2(2)=\overline{\mathfrak M}(2),
]

and (\psi_1(\beta)) and (\psi_2(\beta)) are monotone functions of (\beta); hence it follows that the set of all such values (L(\alpha)) has the cardinality of the continuum. All these values are situated in an arbitrarily small neighborhood of the number

[
\frac{1}{A}=\frac{22}{65+9\sqrt3},
]

whence the theorem follows.

Theorem 4. The minimal point ((N^2+4N)^{-1/2}) of the set (M_L(N)) is a point of the set (M_L(N+1)) for (N\geq 3).

Received
26 VI 1957

References

(^{1}) J. F. Koksma, Diophantische Approximationen, Berlin, 1936. (^{2}) A. G. Kogonia, DAN, 78, No. 4, 637 (1951).