Quantitative metric theory of continued fractions

Transcription

1 Proc. Indian Acad. Sci. Math. Sci. Vol. 6, No., May 06, pp c Indian Academy of Sciences Quantitative metric theory of continued fractions J HANČL, A HADDLEY, P LERTCHOOSAKUL 3 and R NAIR, Department of Mathematics and Centre for Excellence IT4Innovation, Division of UO, Institute for Research and Applications of Fuzzy Modeling, University of Ostrava, 30. dubna, Ostrava, Czech Republic Mathematical Sciences, The University of Liverpool, Peach Street, Liverpool L69 7ZL, United Kingdom 3 Institute of Mathematics, Polish Academy of Sciences, ul Śniadechich 8, Warsaw, Poland * Corresponding author. jaroslav.hancl@seznam.cz; ajassova@liverpool.ac.uk; lertchoosakul@hotmail.com; nair@liv.ac.uk MS received 8 April 04; revised 9 June 05 Abstract. Quantitative versions of the central results of the metric theory of continued fractions were given primarily by C. De Vroedt. In this paper we give improvements of the bounds involved. For a real number x,let x = c 0 +. c + c + c 3 + c A sample result we prove is that given ǫ>0, c x c n x n = k= + almost everywhere with respect to the Lebesgue measure. logk log +o n log n 3 log log n +ǫ kk + Keywords. Continued fractions; ergodic averages; metric theory of numbers. Mathematics Subject Classification. Primary: K50; Secondary: 8D99.. Introduction In this paper, we use a quantitative L -ergodic theorem to study the metrical theory of the regular continued fraction expansion of real number. Here and throughout the rest of the paper, by a dynamical system,β,μ,twe mean a set, together with a σ -algebra β of subsets of, a probability measure μ on the measurable space,β and a measurable self-map T of that is also measure-preserving. By this we mean that if given an element A of β and if we set T A ={x : Tx A}, then μa = μt A. Wesaya dynamical system is ergodic if T A = A for some A in β means that μa is either zero 67

2 68 JHančl et al. or one in value. For β-measurable f,let f denote the L norm f dμ. Asa standard, for two functions f and g, wesayfx= Ogx if there exists C>0such fx that fx C gx. Wesayfx = ogx, asx, if lim x gx = 0. The central tool in this note is the following theorem. Theorem. Let,β,μ,Tdenote an ergodic dynamical system. Suppose that N N ft n x f dμ B, N for B>0 and N =,,... Then given ǫ>0, we have N ft n x = N for μ almost all x. For a real number x, let x = c 0 + c + c + c 3 + c log N 3 log log N +ǫ f dμ + o, denote its regular continued fraction expansion, which is also written more compactly as [c 0 ;c,c,...]. Thetermsc 0,c,... are called the partial quotients of the continued fraction expansion and the sequence of rational truncates [c 0 ;c,...,c n ]= p n q n, n=,,... are called the convergents of the continued fraction expansion. In this paper, we use Theorem to study the metric theory of continued fractions. In particular, we refine results of DeVroedt [, 3]. For more historic background, see [4, 5] and for basic background on continued fractions, see [6]. For a real number y, let{y} denote its fractional part. We now consider the particular ergodic properties of the Gauss map, defined on [0, ] by {{ } x, if x = 0, Tx= 0, if x = 0. Notice that c n x = c n T x n =,,...Let,β,μ,T denote the dynamical system where denotes [0, ], β is the σ -algebra of Borel sets on, μ = γ is the measure on,β defined for any A in β by γa= log A dx x +, N

3 Quantitative metric theory of continued fractions 69 and T is the Gauss map. Note that,β,μ,tis ergodic see [] for details. The ergodic properties of the Gauss dynamical system,β,μ,tare not quite enough to carry out this investigation. We also need ergodic theoretic information about its natural extension which we now describe. Let = [0, \ Q [0, ]. Nowletβ be the σ -algebra of Borel subsets of and let γ be the probability measure on the measurable space,β defined by γ A = log A dxdy + xy. Also define the map T x,y = Tx, [ x ]+y. Then the map T preserves the measure γ and the dynamical system,β,γ,t is ergodic, which we call the natural extension of the Gauss dynamical system see [8] for details. Our results are obtained by applying Theorem to the maps T and T.In particular we need the following theorem from [8] see [] for a definition of the natural extension. In, we prove Theorem. In 3 we state and derive our results about the metric theory of continued fractions.. Proof of Theorem To prove Theorem, we need the following lemma. Lemma.. Suppose,β,μ,Tis an ergodic dynamical system and suppose f : R is μ-integrable with f dμ = 0. Also suppose, for non-negative integer k, that k ft n x dμ B k, for B>0. Then also for k 0, j max ft n x dμ = Ok k. j k Proof of Lemma.. Let and let FM,N,x= N n=m ft n x, 0 M<N m n f,x = max F0,l,x, n. l n

4 70 JHančl et al. Suppose j< K+. Note that any natural number j can be written uniquely in the form j = a + + a h, where a i depends on j with a > >a h 0. With our bound on j, it follows that h = hj < K +. With this notation, we have h F0,j,x= Fl i,l i+,x, where l 0 = 0, l i = l i j = a + + a i for i h, and l h = j. Note that l i = 0 mod a i for all i and all choices of j. By the Cauchy Schwarz inequality, h F0,j,x K + Fl i,l i+,x, j =,,...,h. Also, regardless of j or of the particular sequence of l i pertaining to j, wehave h Fl i,l i+,x i= K+ p= p ν =0 ν even Fν K+ p,ν+ K+ p,x. This is because every term on the left occurs as a term on the right. Integrating over, and using the premise that T is measure-preserving, we obtain that m K f,xdμ K + K+ p= K+ = K + p p= p ν =0 ν even By the hypothesis of this lemma, the last quantity is no more than F0, K+ p,x dμ F0, K+ p,x dμ. K+ K+ K + p B K+ p = BK + p K+ p p= p= BC K + K C K. K for some positive constants C,C. This completes the proof of Lemma.. We now complete the proof of Theorem. Given ǫ>0, we set gn = BN log N 3 log log N +ǫ and set E ǫ = { x : lim sup N N ftn x N f dμ } > 0. gn

5 Quantitative metric theory of continued fractions 7 To prove Theorem, we need to show μe ǫ = 0. Set { l } A k = x : max ft n x l f dμ 4 k l<4 k >log k ǫ/ g4 k. One checks easily that E ǫ r= k=r A k. By the Borel Cantelli lemma, we need to show k= μa k <. Note that μa k log k ǫ g4 k max l A k 4 k l<4 k By the premise of Theorem, for all k, k ft n x k f dμ dμ B k. Thus, by Lemma., there exists C > 0 such that max l 0 l k ft n x l Hence, there exists C 0 > 0 such that ft n x l f dμ dμ C k k. μa k log k ǫ 4 k k log 4 3 logk log 4 +ǫ C 0 k 4 k f dμ dμ. and so μa k = O klog k +ǫ. It follows that k= μa k < and so Theorem is proved. 3. Mixing properties of continued fraction maps Let B k denote the σ -algebra of the subsets of generated by the sequence of functions a α,...,a k α and letbk denote the σ -algebra generated by the sequence of functions a k α,a k+ α,... For a Borel probability measure μ on [0, and for each natural number n, set ψ μ n = sup μa B μaμb, where the supremum is taken over all sets A B k and B B k+n, with μaμb = 0for all natural numbers k. We call the sequence a n n N ψ-mixing if ψn = o. In [4], it is shown that ψ γ n ρ n where ρ 0, 0.8. It then follows readily that ψ γ n ρ n.

6 7 JHančl et al. From Chapter of [5], we have Lemma 3.. Let at n x n N, be a ψ-mixing for dynamical system,β,μ,tand suppose n N ψn <. Also suppose for a function f with domain containing the range of a, that f ax dμ<. Then the series σ = f axdμ { fax + n N faxdμ } dμ, faxdμ is absolutely convergent and non-negative and we have N fat n x N n= f dμ faxdμ fat n x = σn + o. Evidently n= ρ n <. We have shown that ψ γ n,ψ γ n = On as n tends to infinity. 4. Statistical properties of continued fractions In this section, we use Theorem to refine classical results of metric theory of continued fractions [, 3, 7, 9, 3]. See also [] for related matter on subsequence averages. This is possible because we use Theorem instead of the classical and well known method of I. S. Gál and J. F. Koksma, which for any ǫ>0gives an error term of order on log n 3 +ǫ. The improvement seems small but is the first in nearly 50 years. For i m = i,...,i m N m, suppose p nα q n α = [i,...,i n ] n =,,... are the convergents of the continued fraction expansion of the real number α, with g.c.d.p n,q n = g.c.d.p n,q n = and p 0 = q 0 =. Let and Ki m = Li m = { pm +p m q m +q m, if m is odd, p m if m is even qm, { pm +p m q m +q m, if m is even, p m if m is odd. qm, We have the following application of Theorem. PROPOSITION 4. Suppose m N and let F : N m R be such that F c x,...,c m xdγx<

7 or equivalently that Let Then Quantitative metric theory of continued fractions 73 i m N m Fi m Li m Ki m <. δ m = log i m N m Fi m log + Ki m + Li m. n Fc k x,...,c k+m x = δ m + on 3 log n log log n +ǫ n k= almost everywhere with respect to Lebesgue measure. Proof. Take fx= Fc x,...,c m x in Lemma 3.. Proposition 4. has a number of consequences, that follow from self-evident choices of m and F, which we collect together in the following proposition. We do, however, need the following lemma from [3]. Lemma 4.. We have p=0 log l+pm+ l+pml+pm+ = log Ŵ m l l+ Ŵ Ŵ l+ m For brevity set n,ǫ = n log n 3 log log n +ǫ n =,,... PROPOSITION 4. Given ǫ>0, we have a n #{k [,n]:c kx = i}= log + ii+ + o n,ǫ; b n #{k [,n]:c kx i}= log i+ i + o n,ǫ; c n #{k [,n]:i c kx j}= log i+j+ ij+ + o n,ǫ; d n #{k [,n]:c kx,...,c k+m x = i m }= log +Ki m +Li m e n #{k [,n]:c kx l mod m}= log Ŵ m l l+ Ŵ and f for p, p = 0 we have logi k= + kk+ log and K p = to the Lebesgue measure. c x p + +c n x p N 0 m Ŵ l+ m m. + o n,ǫ; + o n,ǫ; p = K p + o n,ǫ, where K = t p p t+, almost everywhere with respect To the sequence of strictly stationary random variables a n x n N, defined on the measurable space [0,,B[0, via the natural extension construction we can construct the doubly infinite sequence of random variables a n n N on [0,,B[0, defined μ almost everywhere on [0, for any probability measure μ assigning zero to the

8 74 JHančl et al. measure of [0, \ e.g. μ = γ, where =[0, \Q. We define sl =[al,a l,...] and yl = sl. Evidently sl = s0 Tl and yl = y0 Tl.Alsofora [0,, put s0 a = a and sl a = sl a +a. n Samur [6, 7] showed that n σ = lim log y n i x,y π dxdy log + xy [0, [0,\Q exists and is positive. Theorem applied tot, forǫ>0, gives n ft i x,y = n [0, [0,\Q fx,y dxdy + xy + o n,ǫ for f L γ.letf = χ B denote the characteristic function of the set B B [0,. Now noting that from the definition of s k k=, the fact that T x,y = Tx,, x,y [0, \Q [0,, c y + x that for i, T i x,y = T i x, [c i x,...,c x,c x+y], x,y [0, \Q [0,, and that for i 0, T i x,y = T i x,si, x,y [0, \Q [0,. Theorem gives the following proposition. PROPOSITION 4.3 Given ǫ>0, we have n log yn n = n n log s i = π log + o n,ǫ almost everywhere with respect to the Lebesgue measure on [0, \Q [0,. We also have the following result. PROPOSITION 4.4 Given ǫ>0, for any a [0,, n log yi = n n n log s a i = π log + o n,ǫ almost everywhere with respect to the Lebesgue measure on [0, \Q.

9 Quantitative metric theory of continued fractions 75 Proof. By the mean value theorem we can see that logx logy x y minx,y for any 0 < x,y with x = y. Now note that if F m m= denotes the Fibonacci sequence, then 0 < Lim Ki m, F m F m for any interval Ii m =[0, \QKi m,li m i m N m and k N, as follows from basic properties of continued fractions. For a [0,, wehave log s m log sm a max, = og k, F m F m where g denotes the golden ratio, as m tends to for all x,y [0, \Q [0,. The proof of Proposition 4.3 is complete. In the case a = 0, we have s 0 k = q k q k k so we get the following Proposition. PROPOSITION 4.5 Given ǫ>0, we have q n n x = e π log + o n,ǫ almost everywhere with respect to the Lebesgue measure on [0, \Q. PROPOSITION 4.6 F m F m Given ǫ>0, we have n log x p n = π 6log + o n,ǫ q n almost everywhere with respect to the Lebesgue measure on [0, \Q. Proof. Note the classical inequality from Proposition 4.4. PROPOSITION 4.7 < qn+ x p n q n <. The proposition now follows qn Let Ia,...,a n ={x [0, : c x = a,...,c n x = a n }. Given ǫ>0, we have n log λia,a,...,a n = π 6log + o n,ǫ almost everywhere with respect to the Lebesgue measure on [0, \Q. Proof. Let λ denote the Lebesgue measure, and let Ic,...,c n be a cylinder in [0,. Recall that s n = q n q n and that λic,...,c n = q n q n +q n. This means that log λic,...,c n = logq n logs n + for n N. Since s n [0,, the proposition follows from Proposition 4.5.

10 76 JHančl et al. We also have the following proposition. PROPOSITION 4.8 Given ǫ>0, we have p n n x = x n e π log + o n,ǫ almost everywhere with respect to the Lebesgue measure on [0, \Q. Proof. This proposition follows from the inequality p n x n xqn x n F n+ F n n n for all x [0, \Q and n N, which is easily checked. Acknowledgement The authors would like to thank the referee for very detailed comments, which substantially improved the presentation of this paper. The first author JH was supported by Grant No. P0//35. References [] Cornfeld I P, Formin S V and Sinai Ya G, Ergodic Theory 98 Springer Verlag [] De Vroedt C, Measure-theoretic investigation concerning continued fractions, Indag. Math [3] De Vroedt C, Metrical problems concerning continued fractions, Compositio Math [4] Doeblin W, Remarques sur la théorie métrique des fractions continues, Compositio Math [5] Doukhan P, Mixing. Properties and Examples, Lect. Notes. in Statist New York: Springer-Verlag [6] Hardy G H and Wright E, An Introduction to Number Theory 979 Oxford University Press [7] Iosifescu M and Kraaikamp C, Metrical theory of continued fractions 00 Kluwer Academic Publishers [8] Ito S, Nakada H and Tanaka S, On the invariant measure for the transformation associated with some real continued fractions, Keio Engineering Reports [9] Khinchin A Ya, Continued Fractions Chicago: Univ. Chicago Press Translation of the 3rd Russian edition [0] Koksma J F, Diophantische Approximationen 936 Berlin: Julius Springer [] Lévy P, Sur les lois de probabilité dont dépendent les quotients complets and incomplets d une fraction continue, Bull. Soc. Math. France [] Nair R, On the metrical theory of continued fractions, Proc. Amer. Math. Soc [3] Nolte V N, Some probabilistic results on the convergents of continued fractions, Indag. Math. NS

11 Quantitative metric theory of continued fractions 77 [4] Philipp W, Limit theorems for sums of partial quotients of continued fractions, Mh. Math [5] Ryll-Nardzewski C, On the ergodic theorems II, Ergodic Theory of Continued Fractions, Studia Math [6] Samur J D, On some limit theorems for continued fractions, Trans. Amer. Math. Soc [7] Samur J D, Some remarks on a probability limit theorem for continued fractions, Trans. Amer. Math. Soc COMMUNICATING EDITOR: Parameswaran Sankaran