Quotients Of Continued Fraction Expansion Research Articles

The distribution of the large partial quotients in continued fraction expansions

The distribution of the large partial quotients in continued fraction expansions

SOME RESULTS ON THE LARGEST PARTIAL QUOTIENT IN CONTINUED FRACTIONS

Let [Formula: see text] be a function satisfying [Formula: see text] as [Formula: see text]. Write [Formula: see text] where [Formula: see text] denotes the largest partial quotient among the first [Formula: see text] terms in the continued fraction expansion of [Formula: see text]. We prove that [Formula: see text] has full Hausdorff dimension for a large class of functions [Formula: see text], which strengthens the result of [L. Fang and J. Liu, On the largest partial quotients in continued fraction expansions, Fractals 29 (2021) 2150099.].

On the relative growth rate of the product of consecutive partial quotients in continued fraction expansions of Laurent series

Let Fq((z−1)) be the field of formal Laurent series and I be the valuation ideal of Fq((z−1)). For any x∈I, let x=[A1(x),A2(x),…] be its continued fraction expansion and let Pn(x)/Qn(x) be the n-th convergents of x. Given m≥1, for any α>0, we defineDm(α):={x∈I:∏i=n+1n+m‖Ai(x)‖≥‖Qn(x)‖α for ultimatelyn∈N} andFm(α):={x∈I:∏i=n+1n+m‖Ai(x)‖≥‖Qn(x)‖α for infinitely manyn∈N}, where ‖⋅‖ represents the norm of a polynomial in Fq((z−1)). In this paper, we study the size of the above sets from the perspective of Hausdorff dimension.

Slow increasing functions and the largest partial quotients in continued fraction expansions

AbstractLet 0 ⩽ α ⩽ ∞ and ψ be a positive function defined on (0, ∞). In this paper, we will study the level sets L(α, {ψ(n)}), B(α, {ψ(n)}) and T(α, {ψ(n)}) which are related respectively to the sequence of the largest digits among the first n partial quotients {Ln(x)}n≥1, the increasing sequence of the largest partial quotients {Bn(x)}n⩾1 and the sequence of successive occurrences of the largest partial quotients {Tn(x)}n⩾1 in the continued fraction expansion of x ∈ [0,1) ∩ ℚc. Under suitable assumptions of the function ψ, we will prove that the sets L(α, {ψ(n)}), B(α, {ψ(n)}) and T(α, {ψ(n)}) are all of full Hausdorff dimensions for any 0 ⩽ α ⩽ ∞. These results complement some limit theorems given by J. Galambos [4] and D. Barbolosi and C. Faivre [1].

The relative growth rate of the largest partial quotient to the sum of partial quotients in continued fraction expansions

In this paper, we mainly study the growth rate of the largest partial quotient relative to the growth rate of the sum of partial quotients for the continued fraction expansion of an irrational number. We establish that the Hausdorff dimension of the set of points with a given relative growth rate is full.

Subexponentially increasing sums of partial quotients in continued fraction expansions

AbstractWe investigate from a multifractal analysis point of view the increasing rate of the sums of partial quotients $S_{n}(x)=\sum_{j=1}^n a_{j}(x)$, where x = [a1(x), a2(x), . . .] is the continued fraction expansion of an irrational x ∈ (0, 1). Precisely, for an increasing function ϕ : $\mathbb{N}$ → $\mathbb{N}$, one is interested in the Hausdorff dimension of the set E_\varphi = \left\{x\in (0,1): \lim_{n\to\infty} \frac {S_n(x)} {\varphi(n)} =1\right\}. Several cases are solved by Iommi and Jordan, Wu and Xu, and Xu. We attack the remaining subexponential case exp(nγ), γ ∈ [1/2, 1). We show that when γ ∈ [1/2, 1), Eϕ has Hausdorff dimension 1/2. Thus, surprisingly, the dimension has a jump from 1 to 1/2 at ϕ(n) = exp(n1/2). In a similar way, the distribution of the largest partial quotient is also studied.

Differentiability of the Minkowski question mark function

We get necessary and sufficient conditions for the derivative of the Minkowski question mark function ?(x) to be equal to zero or infinity. These conditions are formulated in terms of sums Sx(t)=a1+⋯+at of partial quotients of continued fraction expansion to x=[0;a1,…,at]. In particular we prove that if there exists C such that Sx(t)⩽κ1t+logtlog2+C with κ1=2log1+52log2=1.388+, then ?′(x) exists and ?′(x)=+∞. Another result is as follows. Assume that there exists a constant C such that Sx(t)⩾κ2t−C with κ2=4log5+292−5log(2+5)log5+292−log(2+5)−log2=4.401+. Then ?′(x) exists and ?′(x)=0. We show that our conditions on the sum Sx(t) are the best possible. Our results improve upon earlier theorems by Paradis, Viader, Bibiloni and Dushistova, Moshchevitin.

On sums of degrees of the partial quotients in continued fraction expansions of Laurent series

For any formal Laurent series x = ∑ n = v ∞ c n z − n with coefficients c n lying in some given finite field, let x = [ a 0 ( x ) ; a 1 ( x ) , a 2 ( x ) , … ] be its continued fraction expansion. It is known that, with respect to the Haar measure, almost surely, the sum of degrees of partial quotients deg a 1 ( x ) + ⋯ + deg a n ( x ) grows linearly. In this note, we quantify the exceptional sets of points with faster growth orders than linear ones by their Hausdorff dimension, which covers an earlier result by J. Wu.

On the distribution for sums of partial quotients in continued fraction expansions

Let x ∊ [0, 1) and [a1(x), a2(x), …] be the continued fraction expansion of x. For any n ⩾ 1, write . Khintchine (1935 Compos. Math. 1 361–82) proved that converges in measure to with respect to , where denotes the one dimensional Lebesgue measure. Philipp (1988 Monatsh. Math. 105 195–206) showed that there is not a reasonable normalizing sequence such that a strong law of large numbers is satisfied. In this paper, we show that for any α ⩾ 0, the set is of Hausdorff dimension 1. Furthermore, we prove that the Hausdorff dimension of the set consisting of reals whose sums of partial quotients grow at a given polynomial rate is 1.

On sums of partial quotients in continued fraction expansions

Assume x ∊ [0, 1) taking on its continued fraction expansion as [a1(x), a2(x), …]. For any n ⩾ 1, write . Khintchine (1935 Compos. Math. 1 361–82) proved that Sn(x)/(n log n) converges in measure to 1/log 2 with respect to , where denotes the one-dimensional Lebesgue measure. Philipp (1988 Monatsh. Math. 105 195–206) showed that {an(x), n ⩾ 1} cannot satisfy a strong law of large numbers for any reasonably growing norming sequence. In (Wu and Xu 2008 Preprint), we discussed the sets of continued fractions whose sums of partial quotients tend to infinity with the polynomial growth rate. In this paper, we consider the sets of continued fractions whose sums of partial quotients tend to infinity exponentially and doubly exponentially. The Hausdorff dimensions of such sets are determined.

Beta-expansion and continued fraction expansion over formal Laurent series

Let x ∈ I be an irrational element and n ⩾ 1 , where I is the unit disc in the field of formal Laurent series F ( ( X −1 ) ) , we denote by k n ( x ) the number of exact partial quotients in continued fraction expansion of x, given by the first n digits in the β-expansion of x, both expansions are based on F ( ( X −1 ) ) . We obtain that lim inf n → + ∞ k n ( x ) n = deg β 2 Q * ( x ) , lim sup n → + ∞ k n ( x ) n = deg β 2 Q * ( x ) , where Q * ( x ) , Q * ( x ) are the upper and lower constants of x, respectively. Also, a central limit theorem and an iterated logarithm law for { k n ( x ) } n ⩾ 1 are established.