Numbers badly approximate by fractions with prime denominator

1995 ◽  
Vol 118 (1) ◽  
pp. 1-5 ◽  
Author(s):  
Glyn Harman
Keyword(s):  
Continued Fraction ◽  
Arithmetic Progressions ◽  
The Other ◽  
Strong Result ◽  
Infinitely Many Solutions ◽  
Continued Fraction Expansion ◽  
Primes In Arithmetic Progressions ◽  
Prime Variable ◽  
Image Position ◽  
Almost All

We write ‖x‖ to denote the least distance from x to an integer, and write p for a prime variable. Duffin and Schaeffer [l] showed that for almost all real α the inequalityhas infinitely many solutions if and only ifdiverges. Thus f(x) = (x log log (10x))−1 is a suitable choice to obtain infinitely many solutions for almost all α. It has been shown [2] that for all real irrational α there are infinitely many solutions to (1) with f(p) = p−/13. We will show elsewhere that the exponent can be increased to 7/22. A very strong result on primes in arithmetic progressions (far stronger than anything within reach at the present time) would lead to an improvement on this result. On the other hand, it is very easy to find irrational a such that no convergent to its continued fraction expansion has prime denominator (for example (45– √10)/186 does not even have a square-free denominator in its continued fraction expansion, since the denominators are alternately divisible by 4 and 9).

scholarly journals Diophantine approximation by continued fractions

1991 ◽  
Vol 51 (2) ◽  
pp. 324-330 ◽  
Author(s):  
Jingcheng Tong
Keyword(s):  
Diophantine Approximation ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Irrational Number ◽  
Continued Fraction Expansion ◽  
Image Position

AbstractLet ξ be an irrational number with simple continued fraction expansion be its ith convergent. Let Mi = [ai+1,…, a1]+ [0; ai+2, ai+3,…]. In this paper we prove that Mn−1 < r and Mn R imply which generalizes a previous result of the author.

The distribution of the largest digit in continued fraction expansions

2009 ◽  
Vol 146 (1) ◽  
pp. 207-212 ◽  
Author(s):  
JUN WU ◽  
JIAN XU
Keyword(s):  
Hausdorff Dimension ◽  
Continued Fraction ◽  
Continued Fraction Expansion ◽  
Image Position ◽  
Continued Fraction Expansions

AbstractLet [a1(x), a2(x), . . .] be the continued fraction expansion of x ∈ [0,1). Write Tn(x)=max{ak(x):1 ≤ k ≤ n}. Philipp [6] proved that Okano [5] showed that for any k ≥ 2, there exists x ∈ [0, 1) such that T(x)=1/log k. In this paper we show that, for any α ≥ 0, the set is of Hausdorff dimension 1.

Almost-primes in arithmetic progressions and short intervals

1978 ◽  
Vol 83 (3) ◽  
pp. 357-375 ◽  
Author(s):  
D. R. Heath-Brown
Keyword(s):  
Prime Numbers ◽  
Arithmetic Progressions ◽  
Short Intervals ◽  
Almost Primes ◽  
Primes In Arithmetic Progressions ◽  
Image Position

In this paper we shall investigate the occurrence of almost-primes in arithmetic progressions and in short intervals. These problems correspond to two well-known conjectures concerning prime numbers. The first conjecture is that, if (l, k) = 1, there exists a prime p satisfying

scholarly journals Bernoulliness of when is an irrational rotation: towards an explicit isomorphism

2020 ◽  
pp. 1-26
Author(s):  
CHRISTOPHE LEURIDAN
Keyword(s):  
Continued Fraction ◽  
Bernoulli Shift ◽  
Irrational Number ◽  
Unit Interval ◽  
Positive Answer ◽  
Short Paper ◽  
Continued Fraction Expansion ◽  
Skew Products ◽  
Image Position ◽  
Open Question

Let $\unicode[STIX]{x1D703}$ be an irrational real number. The map $T_{\unicode[STIX]{x1D703}}:y\mapsto (y+\unicode[STIX]{x1D703})\!\hspace{0.6em}{\rm mod}\hspace{0.2em}1$ from the unit interval $\mathbf{I}= [\!0,1\![$ (endowed with the Lebesgue measure) to itself is ergodic. In a short paper [Parry, Automorphisms of the Bernoulli endomorphism and a class of skew-products. Ergod. Th. & Dynam. Sys.16 (1996), 519–529] published in 1996, Parry provided an explicit isomorphism between the measure-preserving map $[T_{\unicode[STIX]{x1D703}},\text{Id}]$ and the unilateral dyadic Bernoulli shift when $\unicode[STIX]{x1D703}$ is extremely well approximated by the rational numbers, namely, if $$\begin{eqnarray}\inf _{q\geq 1}q^{4}4^{q^{2}}~\text{dist}(\unicode[STIX]{x1D703},q^{-1}\mathbb{Z})=0.\end{eqnarray}$$ A few years later, Hoffman and Rudolph [Uniform endomorphisms which are isomorphic to a Bernoulli shift. Ann. of Math. (2)156 (2002), 79–101] showed that for every irrational number, the measure-preserving map $[T_{\unicode[STIX]{x1D703}},\text{Id}]$ is isomorphic to the unilateral dyadic Bernoulli shift. Their proof is not constructive. In the present paper, we relax notably Parry’s condition on $\unicode[STIX]{x1D703}$ : the explicit map provided by Parry’s method is an isomorphism between the map $[T_{\unicode[STIX]{x1D703}},\text{Id}]$ and the unilateral dyadic Bernoulli shift whenever $$\begin{eqnarray}\inf _{q\geq 1}q^{4}~\text{dist}(\unicode[STIX]{x1D703},q^{-1}\mathbb{Z})=0.\end{eqnarray}$$ This condition can be relaxed again into $$\begin{eqnarray}\inf _{n\geq 1}q_{n}^{3}~(a_{1}+\cdots +a_{n})~|q_{n}\unicode[STIX]{x1D703}-p_{n}|<+\infty ,\end{eqnarray}$$ where $[0;a_{1},a_{2},\ldots ]$ is the continued fraction expansion and $(p_{n}/q_{n})_{n\geq 0}$ the sequence of convergents of $\Vert \unicode[STIX]{x1D703}\Vert :=\text{dist}(\unicode[STIX]{x1D703},\mathbb{Z})$ . Whether Parry’s map is an isomorphism for every $\unicode[STIX]{x1D703}$ or not is still an open question, although we expect a positive answer.

scholarly journals On the metric theory of the optimal continued fraction expansion

1997 ◽  
Vol 56 (1) ◽  
pp. 69-79
Author(s):  
R. Nair
Keyword(s):  
Lebesgue Measure ◽  
Continued Fraction ◽  
Natural Numbers ◽  
Real Numbers ◽  
Continued Fraction Expansion ◽  
Image Position ◽  
Special Case

Suppose kn denotes either φ(n) or φ(rn) (n = 1, 2, …) where the polynomial φ maps the natural numbers to themselves and rk denotes the kth rational prime. Let denote the sequence of convergents to a real numbers x for the optimal continued fraction expansion. Define the sequence of approximation constants byIn this paper we study the behaviour of the sequence for all most all x with respect to Lebesgue measure. In the special case where kn = n (n = 1, 2, …) these results are due to Bosma and Kraaikamp.

Some properties of the continued fraction expansion of (m/n) e1/q

1970 ◽  
Vol 67 (1) ◽  
pp. 67-74 ◽  
Author(s):  
K. R. Matthews ◽  
R. F. C. Walters
Keyword(s):  
Continued Fractions ◽  
Continued Fraction ◽  
Continued Fraction Expansion ◽  
Image Position ◽  
Positive Integers

Introduction. Continued fractions of the form are called Hurwitzian if b1, …, bh, are positive integers, ƒ1(x), …, ƒk(x) are polynomials with rational coefficients which take positive integral values for x = 0, 1, 2, …, and at least one of the polynomials is not constant. f1(x), …, fk(x) are said to form a quasi-period.

Metric Diophantine approximation with two restricted variables I. Two square-free integers, or integers in arithmetic progressions

1988 ◽  
Vol 103 (2) ◽  
pp. 197-206 ◽  
Author(s):  
Glyn Harman
Keyword(s):  
Positive Integer ◽  
Lebesgue Measure ◽  
Diophantine Approximation ◽  
Positive Function ◽  
Arithmetic Progressions ◽  
Integer Variable ◽  
Number Of Solutions ◽  
Metric Diophantine Approximation ◽  
Image Position ◽  
Almost All

In this paper, together with [7] and [8], we shall be concerned with estimating the number of solutions of the inequalityfor almost all α (in the sense of Lebesgue measure on Iℝ), where , and both m and n are restricted to sets of number-theoretic interest. Our aim is to prove results analogous to the following theorem (an improvement given in [2] of an earlier result of Khintchine [10]) and its quantitative developments (for example, see [11, 12,6]):Let ψ(n) be a non-increasing positive function of a positive integer variable n. Then the inequality (1·1) has infinitely many, or only finitely many, solutions in integers to, n(n > 0) for almost all real α, according to whether the sumdiverges, or converges, respectively.

scholarly journals On the Quantitative Metric Theory of Continued Fractions in Positive Characteristic

2018 ◽  
Vol 61 (1) ◽  
pp. 283-293
Author(s):  
Poj Lertchoosakul ◽  
Radhakrishnan Nair
Keyword(s):  
Finite Field ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Laurent Series ◽  
Positive Characteristic ◽  
Continued Fraction Expansion ◽  
Formal Laurent Series ◽  
Almost Everywhere ◽  
Partial Quotients ◽  
Image Position

AbstractLet 𝔽q be the finite field of q elements. An analogue of the regular continued fraction expansion for an element α in the field of formal Laurent series over 𝔽q is given uniquely by $$\alpha = A_0(\alpha ) + \displaystyle{1 \over {A_1(\alpha ) + \displaystyle{1 \over {A_2(\alpha ) + \ddots }}}},$$ where $(A_{n}(\alpha))_{n=0}^{\infty}$ is a sequence of polynomials with coefficients in 𝔽q such that deg(An(α)) ⩾ 1 for all n ⩾ 1. In this paper, we provide quantitative versions of metrical results regarding averages of partial quotients. A sample result we prove is that, given any ϵ > 0, we have $$\vert A_1(\alpha ) \ldots A_N(\alpha )\vert ^{1/N} = q^{q/(q - 1)} + o(N^{ - 1/2}(\log N)^{3/2 + {\rm \epsilon }})$$ for almost everywhere α with respect to Haar measure.

On the Lower Range of Perron's Modular Function

1969 ◽  
Vol 21 ◽  
pp. 808-816 ◽  
Author(s):  
J. R. Kinney ◽  
T. S. Pitcher
Keyword(s):  
Finite Number ◽  
Continued Fraction ◽  
Modular Function ◽  
Finite Collection ◽  
Continued Fraction Expansion ◽  
Lower Range ◽  
Image Position ◽  
Positive Integers

The modular function Mwas introduced by Perron in (6). M(ξ) (for irrational ξ) is denned by the property that the inequalityis satisfied by an infinity of relatively prime pairs (p, q)for positive d,but by at most a finite number of such pairs for negative d.We will writefor the continued fraction expansion of ξ ∈ (0, 1) and for any finite collection y1,…, ykof positive integers we will writeIt is known (see 6) thatWhere

scholarly journals Closed form expressions for two harmonic continued fractions

2017 ◽  
Vol 101 (552) ◽  
pp. 439-448 ◽  
Author(s):  
Martin Bunder ◽  
Joseph Tonien
Keyword(s):  
Closed Form ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Interested Reader ◽  
Continued Fraction Expansion ◽  
Decimal Expansion ◽  
Irrational Numbers ◽  
Image Position ◽  
Mathematical Constants

A continued fraction is an expression of the formThe expression can continue for ever, in which case it is called aninfinitecontinued fraction, or it can stop after some term, when we call it afinitecontinued fraction. For irrational numbers, a continued fraction expansion often reveals beautiful number patterns which remain obscured in their decimal expansion. The interested reader is referred to [1] for a collection of many interesting continued fractions for famous mathematical constants.

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