On the growth behavior of partial quotients in continued fractions

Archiv der Mathematik - Let $$[a_1(x),a_2(x),a_3(x),\ldots ]$$ be the continued fraction expansion of an irrational number $$x\in (0,1)$$ . It is known that for Lebesgue almost all $$x\in...     

A remark on continued fractions with sequences of partial quotients

Given any infinite set B of positive integers , let τ(B) denote the exponent of convergence of the series . Let E(B) be the set . Hirst [K.E. Hirst, Continued fractions with sequences of partial quotients, Proc. Amer. Math. Soc. 38 (1973) 221-227] proved the inequality and conjectured (see Hirst [K.E. Hirst, Continued fractions with sequences of partial quotients, Proc. Amer. Math. Soc. 38 (1973), p. 225] and Cusick [T.W. Cusick, Hausdorff dimension of sets of continued fractions, Quart. J. Math. Oxford Ser. (2) 41 (1990), p. 278]) that equality holds in general. In [Bao-Wei Wang, Jun Wu, A problem of Hirst on continued fractions with sequences of partial quotients, Bull. London Math. Soc., in press], we gave a positive answer to this conjecture. In this note, we further show that the result in [Bao-Wei Wang, Jun Wu, A problem of Hirst on continued fractions with sequences of partial quotients, Bull. London Math. Soc., in press] is sharp.     

Limit theorems for sums of partial quotients of continued fractions

We prove that the partial quotientsa _j of the regular continued fraction expansion cannot satisfy a strong law of large numbers for any reasonably growing norming sequence, and that thea _j belong to the domain of normal attraction to a stable law with characteristic exponent 1. We also show that thea _j satisfy a central limit theorem if a few of the largest ones are trimmed.In memory of Wilfried Nöbauer     

A problem of Hirst on continued fractions with sequences of partial quotients

Let B denote an infinite sequence of positive integers b₁ <b₂ < ..., and let denote the exponent of convergence ofthe series _{n = 1} 1/b_n; that is, = inf {s 0 : _{n = 1} 1/b_n^s <}. Define E(B) = {x [0, 1]: a_n(x) B (n 1) and a_n(x) asn }. K. E. Hirst [Proc. Amer. Math. Soc. 38 (1973) 221–227]proved the inequality dim_H E(B) /2 and conjectured (see ibid.,p. 225 and [T. W. Cusick, Quart. J. Math. Oxford (2) 41 (1990)p. 278]) that equality holds. In this paper, we give a positiveanswer to this conjecture.     

Explicit continued fractions with expected partial quotient growth

For let be the continued fraction expansion of . Write

We construct some numbers 's with

     

Cantor sets determined by partial quotients of continued fractions of Laurent series

In this paper, two types of general sets determined by partial quotients of continued fractions over the field of formal Laurent series with coefficients from a given finite field are studied. The Hausdorff dimensions of and are determined completely, where A_n(x) denotes the partial quotients in the continued fraction expansion (in case of Laurent series) of x and (n) is a positive valued function defined on natural numbers N.     

Dyadic fractions with small partial quotients

It is proved that ifm is a power of 2, then there exists an odd integera with 1a<m such that all partial quotients in the continued fraction expansion ofa/m are bounded by 3. The upper bound 3 is best possible. Similar results can be shown for powers of other small numbers.     

Continued fractions with bounded partial quotients

This paper gives the exact bound of the continued fraction expansion of when has bounded partial quotients and is a Möbius transformation where all entries are integers.

     

The distribution of the large partial quotients in continued fraction expansions

The existence of large partial quotients destroys many limit theorems in the metric theory of continued fractions. To achieve some variant forms of limit theorems, a common approach mostly used in practice is to discard the largest partial quotient, while this approach works in obtaining limit theorems only when there cannot exist two terms of large partial quotients in a metric sense. Motivated by this, we are led to consider the metric theory of points with at least two large partial quotients...     

The theta group and the continued fraction expansion with even partial quotients

F. Schweiger introduced the continued fraction with even partial quotients. We will show a relation between closed geodesics for the theta group (the subgroup of the modular group generated by z+2 and -1 / z) and the continued fraction with even partial quotients. Using thermodynamic formalism, Tauberian results and the above-mentioned relation, we obtain the asymptotic growth number of closed trajectories for the theta group. Several results for the continued fraction expansion with even partial quotients are obtained; some of these are analogous to those already known for the usual continued fraction expansion related to the modular group, but our proofs are by necessity in general technically more difficult.Supported by The Netherlands Organization for Scientific Research (NWO).     

On the fast Khintchine spectrum in continued fractions

For $x\in [0,1)$ x ∈ [ 0 , 1 ) , let $x=[a_1(x), a_2(x),\ldots ]$ x = [ a 1 ( x ) , a 2 ( x ) , ... ] be its continued fraction expansion with partial quotients $\{a_n(x), n\ge 1\}$ { a n ( x ) , n ≥ 1 } . Let $\psi : \mathbb{N } \rightarrow \mathbb{N }$ ψ : N → N be a function with $\psi (n)/n\rightarrow \infty $ ψ ( n ) / n → ∞ as $n\rightarrow \infty $ n → ∞ . In this note, the fast Khintchine spectrum, i.e., the Hausdorff dimension of the set $$\begin{aligned} E(\psi ):=\left\{ x\in [0,1): \lim _{n\rightarrow \infty }\frac{1}{\psi (n)}\sum _{j=1}^n\log a_j(x)=1\right\} \end{aligned}$$ E ( ψ ) : = x ∈ [ 0 , 1 ) : lim n → ∞ 1 ψ ( n ) ∑ j = 1 n log a j ( x ) = 1 is completely determined without any extra condition on $\psi $ ψ . This fills a gap of the former work in Fan et al. (Ergod Theor Dyn Syst 29:73–109, 2009).     

Rational functions with partial quotients of small degree in their continued fraction expansion

For a rational functionf/g=f(x)/g(x) over a fieldF with ged (f,g)=1 and deg (g)1 letK(f/g) be the maximum degree of the partial quotients in the continued fraction expansion off/. ForfF[x] with deg (f)=k1 andf(O)O putL(f)=K(f(x)/x ^k ). It is shown by an explicit construction that for every integerb with 1bk there exists anf withL(f)=b. IfF=F ₂, the binary field, then for everyk there is exactly onefF ₂[x] with deg (f)=k,f(O)O, andL(f)=1. IfF _q is the finite field withq elements andgF _q [x] is monic of degreek1, then there exists a monic irreduciblefF _q [x] with deg (f)=k, gcd (f,g)=1, andK(f/g)<2+2 (logk)/logq, where the caseq=k=2 andg(x)=x ²+x+1 is excluded. An analogous existence theorem is also shown for primitive polynomials over finite fields. These results have applications to pseudorandom number generation.     

Rational functions over finite fields having continued fraction expansions with linear partial quotients

Let F be a finite field with q elements and let g be a polynomial in F[X] with positive degree less than or equal to q/2. We prove that there exists a polynomial f∈F[X], coprime to g and of degree less than g, such that all of the partial quotients in the continued fraction of g/f have degree 1. This result, bounding the size of the partial quotients, is related to a function field equivalent of Zaremba's conjecture and improves on a result of Blackburn [S.R. Blackburn, Orthogonal sequences of polynomials over arbitrary fields, J. Number Theory 6 (1998) 99-111]. If we further require g to be irreducible then we can loosen the degree restriction on g to deg(g)?q.     

Upper and lower fast Khintchine spectra in continued fractions

For an irrational number \(x\in [0,1)\), let \(x=[a_1(x), a_2(x),\ldots ]\) be its continued fraction expansion. Let \(\psi : \mathbb {N} \rightarrow \mathbb {N}\) be a function with \(\psi (n)/n\rightarrow \infty \) as \(n\rightarrow \infty \). The (upper, lower) fast Khintchine spectrum for \(\psi \) is defined as the Hausdorff dimension of the set of numbers \(x\in (0,1)\) for which the (upper, lower) limit of \(\frac{1}{\psi (n)}\sum _{j=1}^n\log a_j(x)\) is equal to 1. The fast Khintchine spectrum was determined by Fan, Liao, Wang, and Wu. We calculate the upper and lower fast Khintchine spectra. These three spectra can be different.