New York Journal of Mathematics

New York J. Math. 3A (1998) 117–124.

On Metric Diophantine Approximation and Subsequence Ergodic Theory

R. Nair

Abstract. Suppose k_{n} denotes either ϕ(n) or ϕ(p_{n}) (n = 1, 2, ⋯) where the polynomial ϕ maps the natural numbers to themselves and p_{k} denotes the k^{th} rational prime. Let ( \frac{r_{n}}{q_{n}})_{n=1}^{∞} denote the sequence of convergents to a real number x and define the the sequence of approximation constants (θ_{n}(x))_{n=1}^{∞} by

θ_{n}(x) = q_{n}^{2} |x − \frac{r_{n}}{q_{n}} | .  (n = 1, 2, ⋯)

In this paper we study the behaviour of the sequence (θ_{k_{n}}(x))_{n=1}^{∞} for almost all x with respect to Lebesgue measure. In the special case where k_{n} = n (n = 1, 2, ⋯) these results are due to W. Bosma, H. Jager and F. Wiedijk.

Contents

1. Introduction 117 2. Basic Ergodic Theory 118 3. Statistical Properties of the Sequence (θ_{n}(x))_{n=1}^{∞} 120 4. Other Sequences Attached to the Continued Fraction Expansion 121 References 123

1. Introduction

In this paper we study the behaviour of the regular continued fraction expansion of a real number

x = c_{0} + \frac{1}{c_{1} + \frac{1}{c_{2} + \frac{1}{c_{3} + \frac{1}{c_{4} ...}}}} ,

Received November 7, 1997. Mathematics Subject Classification. 11K50, 28D99. Key words and phrases. metric diophantine approximation, continued fractions, subsequence

ergodic theorems.

\begin{matrix}c \\ ◯ \end{matrix}1998 State University of New York

ISSN 1076-9803/98

which is also written more compactly as [c_{0}; c_{1}, c_{2}, …]. The terms c_{0}, c_{1}, … are called the partial quotients of the continued fraction expansion and the sequence of rational truncates

[c_{0}; c_{1}, …, c_{n}] = \frac{p_{n}}{q_{n}} ,  (n = 1, 2, …)

are called the convergents of the continued fraction expansion. More particularly recall the inequality

|x − \frac{p_{n}}{q_{n}} | ≤ \frac{1}{q_{n}^{2}} , (1.1)

which is classical and well known [HW]. Clearly if for each natural number n we set

θ_{n}(x) = q_{n}^{2} |x − \frac{p_{n}}{q_{n}} | ,

then for each x the sequence (θ_{n}(x))_{n=1}^{∞} lies in the interval [0, 1]. The distribution for almost all x with respect to Lebesgue measure of the sequence (θ_{n}(x))_{n=1}^{∞} is studied in [BJW]. In this paper extending work in [BJW] we use ergodic theory to study some other functions of this sequence. In Section 2 we collect together some ergodic theoretic prerequisites. In Section 3 we state and prove our main result concerning the distribution of (θ_{n}(x))_{n=1}^{∞} which refines the work in [BJW]. Finally in Section 4 the method of Section 3 is adapted to study some other sequences attached to the continued fraction expansion of x.

2. Basic Ergodic Theory

Here and throughout the rest of the paper by a dynamical system (X, β, μ, T) we mean a set X, together with a σ-algebra β of subsets of X, a probability measure μ on the measurable space (X, β) and a measurable self map T of X that is also measure preserving. By this we mean that if given an element A of β if we set T^{−1}A = {x ∈ X : T x ∈ A} then μ(A) = μ(T^{−1}A). We say a dynamical system is ergodic if T^{−1}A = A for some A in β means that μ(A) is either zero or one in value. We say the dynamical system (X, β, μ, T) is weak mixing (among other equivalent formulations [Wa]) if for each pair of sets A and B in β we have

lim_{N→∞} \frac{1}{N} ∑_{n=1}^{N} |μ(T^{−n}A ∩ B) − μ(A)μ(B)| = 0.

Weak mixing is a strictly stronger condition than ergodicity. A piece of terminology that is becoming increasingly standard is to call a sequence k = (k_{n})_{n=1}^{∞} of nonnegative integers L^{p} good universal if given any dynamical system (X, β, μ, T) and any function f in L^{p}(X, β, μ) it is true that

lim_{N→∞} \frac{1}{N} ∑_{n=1}^{N} f(T^{k_{n}}x) = ℓ_{f}(x),

exists almost everywhere with respect to the measure μ. Here and henceforth, for each real number y let [y] denote the greatest integer less than y and let ⟨y⟩ = y − [y]. The following theorem is a consequence of Theorem 2.3 in [Na2].

Theorem 2.1. Suppose the sequence k = (k_{n})_{n=1}^{∞} of non-negative integers is such that for each irrational number α the sequence (⟨k_{n}α⟩)_{n=1}^{∞} is uniformly distributed modulo one and that for a particular p greater or equal to one that k = (k_{n})_{n=1}^{∞} is L^{p} good universal. Then if the dynamical system (X, β, μ, T) is weak mixing ℓ_{f}(x) = ∫_{X} f(t)dμ(t) almost everywhere with respect to μ.

If k_{n} denotes either ϕ(n) or ϕ(p_{n}) where ϕ denotes any non-constant polynomial mapping the natural numbers to themselves and p_{n} denotes the n^{th} rational prime then k is L^{p} good universal for any p greater than one. See [Bo2] and [Na1] respectively for proofs, and the 1989 Ohio State Ph.D thesis of M. Wierdl for related results. The fact that for each irrational number α the sequence (⟨k_{n}α⟩)_{n=1}^{∞} is uniformly distributed modulo one in both instances are well known classical results. See [We] and [Rh] respectively. Other sequences are known by the author to satisfy the both hypotheses but these results have yet to appear in print [Na3].

We now consider the particular ergodic properties of the Gauss map, defined on [0, 1] by

T x = ⟨\frac{1}{x}⟩ x ≠ 0 ; T 0 = 0.

Notice that c_{n}(x) = c_{n−1}(T x) (n = 1, 2, ⋯). The dynamical system (X, β, μ, T) where X denotes [0, 1], β is the σ-algebra of Borel sets on X, μ is the measure on (X, β) defined for any A in β by

μ(A) = \frac{1}{log 2} ∫_{A} \frac{dx}{x + 1},

and T is the Gauss map is weak mixing. See [CFS] for details. The ergodic properties of the dynamical system (X, β, μ, T) are not quite enough to carry out this investigation. We also need ergodic theoretic information about its natural extention. In particular we need the following theorem from [INT]. See [CFS] for a definition of the natural extention and [S] for other general background.

Theorem 2.2. Let Ω = ([0, 1) \ Q) × [0, 1]. Now let γ be the σ-algebra of Borel subsets of Ω and let ω be the probability measure on the measurable space (Ω, β) defined by

ω(A) = \frac{1}{(log 2)} ∫_{A} \frac{dxdy}{(1 + xy)^{2}} .

Also define the map

T (x, y) = (T x, \frac{1}{[\frac{1}{x}] + y}).

Then the map T preserves the measure ω and the dynamical system (Ω, β, ω, T ) is weak mixing.

Note that

T^{n}(x, y) = (T^{n}x, [0; a_{n}, a_{n−1}, ⋯, a_{2}, a_{1} + y])  (0 ≤ y ≤ 1, n = 1, 2, ⋯)

and in particular

T^{n}(x, 0) = (T^{n}x, \frac{q_{n−1}}{q_{n}}).

3. Statistical Properties of the Sequence (θ_{n}(x))_{n=1}^{∞}

The main result of this paper is the following.

Theorem 3.1. Suppose the sequence of integers k = (k_{n})_{n=1}^{∞} satisfies the hypothesis of Theorem 2.1. Let the function F_{1} : [0, 1] → [0, 1] be defined by F_{1}(z) = \frac{z}{log 2} on [0, \frac{1}{2}] and F_{1}(z) = \frac{1}{log 2}(1 − z + log 2z) on [\frac{1}{2}, 1]. Then

lim_{n→∞} \frac{1}{n}|\{1 ≤ j ≤ n : θ_{k_{j}}(x) ≤ z\}| = F_{1}(z), (3.2)

almost everywhere with respect to Lebesgue measure.

In the special case k_{n} = n (n = 1, 2, …) this result was conjectured by H. W. Lenstra Jr. and proved in [BJW].

Proof of Theorem 3.1. Denote by Ω(c) with c ≥ 1 that part of Ω on or above the hyperbola \frac{1}{x} + y = c. In [K, p. 29] it is noted that

θ_{n}(x) = \frac{1}{(\frac{1}{T^{n}x} + \frac{q_{n−1}}{q_{n}})}  (n = 1, 2, …),

the statement θ_{n}(x) ≤ z for z ∈ [0, 1] is equivalent to the statement that T^{n}(x, 0) ∈ Ω(\frac{1}{z}). It is also readily verified there exists an integer n_{0}(ϵ) such that for all n greater than n_{0}(ϵ) and all y in [0, 1] if

T^{n}(x, y) ∈ Ω(\frac{1}{z} + ϵ)

then

T^{n}(x, 0) ∈ Ω(\frac{1}{z}).

Also if

T^{n}(x, 0) ∈ Ω(\frac{1}{z})

then

T^{n}(x, y) ∈ Ω(\frac{1}{z} − ϵ).

From this it follows that for almost all (x, y) with respect to the measure μ we have

lim_{n→∞} \frac{1}{n}|\{1 ≤ j ≤ n ; T^{k_{j}}(x, y) ∈ Ω(\frac{1}{z} + ϵ)\}|

≤ lim inf_{n→∞} \frac{1}{n}|\{1 ≤ j ≤ n ; T^{k_{j}}(x, 0) ∈ Ω(\frac{1}{z})\}|

≤ lim sup_{n→∞} \frac{1}{n}|\{1 ≤ j ≤ n ; T^{k_{j}}(x, 0) ∈ Ω(\frac{1}{z})\}|

≤ lim_{n→∞} \frac{1}{n}|\{1 ≤ j ≤ n ; T^{k_{j}}(x, y) ∈ Ω(\frac{1}{z} − ϵ)\}|.

Using the fact that k = (k_{n})_{n=1}^{∞} is L^{p} good universal, both limits exist and are μ(Ω(\frac{1}{z} + ϵ)) and μ(Ω(\frac{1}{z} − ϵ)) respectively. Since ϵ is arbitrary the limit (3.2) exists and is equal to μ(Ω(\frac{1}{z})) for almost all x with respect to Lebesgue measure. We straightforwardly verify that μ(Ω(\frac{1}{z})) = F (z). □

Corollary 3.3. Suppose the sequence k = (k_{n})_{n=1}^{∞} satisfies the hypothesis of Theorem 2.1. Then

lim_{n→∞} \frac{1}{n} ∑_{j=1}^{n} θ_{k_{j}}(x) = \frac{1}{4 log 2},

almost everywhere with respect to Lebesgue measure.

Proof. This follows immediately from the fact that the first moment ∫_{0}^{1} zdF_{1}(z) has the value \frac{1}{4 log 2}. □

4. Other Sequences Attached to the Regular Continued Fraction Expansion

Theorem 4.1. Suppose z is in [0, 1] and for irrational x in (0, 1) set Q_{n}(x) = \frac{q_{n−1}(x)}{q_{n}(x)} for each positive integer n. Suppose also that k = (k_{n})_{n=1}^{∞} satisfies the hypothesis of Theorem 2.1. Then

lim_{n→∞} \frac{1}{n}|\{1 ≤ j ≤ n : Q_{k_{j}}(x) ≤ z\}| = F_{2}(z) = \frac{log(1 + z)}{log 2}

almost everywhere with respect to Lebesgue measure.

Proof. Using the fact that k = (k_{n})_{n=1}^{∞} satisfies the hypothesis of Theorem 2.1, we see that

lim_{n→∞} \frac{1}{n}|\{1 ≤ j ≤ n : T^{k_{j}}(x) ≤ z\}| = \frac{1}{log 2} ∫_{0}^{z} \frac{dx}{1 + x} = \frac{log(1 + z)}{log 2}.

Now note that, for a set E in β if \overline{E} denotes \{(x, y) : (y, x) ∈ E\} then μ(E) = μ(\overline{E}) and so (Q_{k_{j}}(x))_{n=1}^{∞} is distributed identically to (T^{k_{j}}x)_{j=1}^{∞} and the theorem follows as a consequence. □

Theorem 4.2. For irrational x in (0, 1) set

r_{n}(x) = \frac{|x − \frac{p_{n}}{q_{n}} |}{|x − \frac{p_{n−1}}{q_{n−1}} |}.  (n = 1, 2, …)

Further for z in [0, 1] let

(4.3) F_{3}(z) = \frac{1}{log 2}(log(1 + z) − \frac{z}{1 + z} log z).

Suppose also that k = (k_{n})_{n=1}^{∞} satisfies the hypothesis of Theorem 2.1. Then

lim_{n→∞} \frac{1}{n}|\{1 ≤ j ≤ n : r_{k_{j}}(x) ≤ z\}| = F_{3}(z),

almost everywhere with respect to Lebesgue measure.

Proof. It follows from the fact that

x − \frac{p_{n}}{q_{n}} = \frac{(−1)^{n}T^{n}x}{q_{n}(q_{n} + q_{n−1}T^{n}x)} (4.4)

[B, pp. 41–42] and the fact that

\frac{1}{T^{n−1}x} = a_{n} + T^{n}x

that r_{n}(x) = \frac{q_{n−1}}{q_{n}} T^{n}x. Arguing as in the proof of Theorem 3.1 we see that F_{3} exists for almost all x and that for z in [0, 1] the value of F_{3}(z) is equal to the μ measure of the part of Ω under the curve xy = z. A simple calculation shows that F_{3} is given by (4.3) as specified. □

Corollary 4.5. Suppose the sequence k = (k_{n})_{n=1}^{∞} satisfies the hypothesis of Theorem 2.1. Then

lim_{n→∞} \frac{1}{n} ∑_{j=1}^{n} r_{k_{j}}(x) = \frac{π^{2}}{12 log 2} − 1,

almost everywhere with respect to Lebesgue measure.

Proof. The limit is ∫_{0}^{1} zdF_{3}(z). □

Another well known inequality is the following

|x − \frac{p_{n}}{q_{n}} | < \frac{1}{q_{n}q_{n+1}}  (n = 1, 2, …)

which motivates the following result.

Theorem 4.6. For each irrational number x in (0, 1) define the function d_{n}(x) for each natural number n by the identity

|x − \frac{p_{n}}{q_{n}} | = \frac{d_{n}(x)}{q_{n}q_{n+1}} . (4.7)

Suppose the sequence k = (k_{n})_{n=1}^{∞} satisfies the hypothesis of Theorem 2.1. Suppose also F_{4} is defined on [0, 1] as F_{4}(z) = 0 if z is in [0, \frac{1}{2}] and

F_{4}(z) = \frac{1}{log 2}(z log z + (1 − z) log(1 − z) + log 2)

if z is in [\frac{1}{2}, 1]. Then

lim_{n→∞} \frac{1}{n}|\{1 ≤ j ≤ n : d_{k_{j}}(x) ≤ z\}| = F_{4}(z),

almost everywhere with respect to Lebesgue measure.

Proof. From (4.4) and (4.7) we readily see that

d_{n}(x) = \frac{1}{1 + \frac{q_{n}}{q_{n−1}} T^{n+1}x}.  (n = 1, 2, …)

Hence F_{4}(z) equals the μ measure of the part of Ω above the curve xy = \frac{1}{z} − 1. Note that for z ≤ \frac{1}{2} this is an empty set. □

Finally in this section we consider the inequality

|x − \frac{p_{n}}{q_{n}} | < \frac{1}{2q_{n}q_{n−1}} .  (n = 1, 2, …)

This is sharper than (1.1) whenever c_{n} = 1. That is for almost all x with frequency 2 − \frac{log 3}{log 2}. See [B] for details. This motivates the following theorem.

Metric Diophantine Approximation 123

Theorem 4.8. For each irrational number x in (0, 1) define the function D_{n}(x) for each natural number n by the identity

|x − \frac{p_{n}}{q_{n}} | = \frac{D_{n}(x)}{q_{n}q_{n−1}} .  (n = 1, 2, …)

Suppose the sequence k = (k_{n})_{n=1}^{∞} satisfies the hypothesis of Theorem 2.1. Suppose F_{5} is defined on [0, 1]

F_{5}(z) = \frac{1}{log 2}(log z − \frac{z}{2} log z − \frac{2 − z}{2} log(2 − z))

if z is in [0, 1]. Then

lim_{n→∞} \frac{1}{n}|\{1 ≤ j ≤ n : D_{k_{j}}(x) ≤ z\}| = F_{5}(z),

almost everywhere with respect to Lebesgue measure.

Proof. It is not difficult to verify that

D_{n}(x) = \frac{2}{(\frac{q_{n}}{q_{n−1}} \frac{1}{T^{n}x} + 1)}.  (n = 1, 2, …)

As earlier in the proof of Theorem 3.1 F_{5}(z) denotes the μ measure of the part of Ω under the hyperbola xy = \frac{z}{2 − z} when z is in [0, 1]. □

Corollary 4.9. Suppose the sequence k = (k_{n})_{n=1}^{∞} satisfies the hypothesis of Theorem 2.1. Then

lim_{n→∞} \frac{1}{n} ∑_{j=1}^{n} D_{k_{j}}(x) = 1 − \frac{1}{2 log 2},

almost everywhere with respect to Lebesgue measure.

Proof. The limit is ∫_{0}^{1} zdF_{5}(z) = 1 − \frac{1}{2 log 2}. □

References

[B] P. Billingsley, Ergodic Theory and Information, John Wiley and Sons, New York, 1965.

[BJW] W. Bosma, H., Jager and F. Wiedijk, Some metrical observations on the approximation by continued fractions, Indag. Math. 45 (1983), 281–299.

[Bo1] J. Bourgain, On the maximal ergodic theorem for certain subsets of the integers, Isr. J. Math. 61 (1988), 39–72.

[Bo2] J. Bourgain, Pointwise ergodic theorems for arithmetic sets, Publ. I.H.E.S. 69 (1989), 5–45.

[CFS] I. P. Cornfeld, S. V. Formin and Ya. G. Sinai, Ergodic Theory, Springer-Verlag, Berlin, 1982.

[HW] G. H. Hardy and E. Wright, An Introduction to Number Theory, Oxford University Press, Oxford, 1979.

[INT] S. Ito, H. Nakada and S. Tanaka, On the invariant measure for the transformation associated with some real continued fractions, Keio Engineering reports 30 (1977), 159–175.

[K] J. F. Koksma, Diophantische Approximationen, Julius Springer, Berlin, 1936.

[Na1] R. Nair, On Polynomials in primes and J. Bourgains’s circle method approach to ergodic theorems II, Studia Math. 105 (1993), 207–233.

[Na2] R. Nair, On Hartman uniform distribution and measures on compact groups, Harmonic Analysis and Hypergroups (M. Anderson, A.I. Singh and K. Ross eds.), Trends in Mathematics, no. 3, Birkhauser, Boston, 1998, pp. 59–75.

[Na3] R. Nair, On uniformly distributed sequences of integers and Poincaré recurrence II, to appear in Indag. Math.

124 R. Nair

[Rh] G. Rhin, Sur la repártition modulo 1 des suites f(p), Acta Arithmetica XXIII (1973), 217–248.

[S] F. Schweiger, Ergodic Theory of Fibered systems and Metric Number Theory, Oxford University Press, Oxford, 1995.

[Wa] P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, no. 79, Springer-Verlag, Berlin, 1982.

[We] H. Weyl, Über die Gleichvertilung von Zahlen mod. Eins, Math. Ann. 77 (1916), 313–361.

Department of Mathematical Sciences, University of Liverpool, P.O. Box 147, Liverpool L69 3BX, U.K.

nair@liv.ac.uk http://www.liv.ac.uk/PureMaths/members/K Nair.html

This paper is available via http://nyjm.albany.edu:8000/j/1998/3A-9.html.