Some metrical theorems in diophantine approximation

1951 ◽  
Vol 47 (1) ◽  
pp. 18-21 ◽  
Author(s):  
J. W. S. Cassels
Keyword(s):  
Real Number ◽  
Diophantine Approximation ◽  
Infinitely Many Solutions ◽  
Real Numbers ◽  
Integer Solutions ◽  
The Difference ◽  
Image Position

Introduction. If ξ is a real number we denote by ∥ ξ ∥ the difference between ξ and the nearest integer, i.e.It is well known (e.g. Koksma (3), I, Satz 4) that if θ1, θ2, …, θn are any real numbers, the inequalityhas infinitely many integer solutions q > 0. In particular, if α is any real number, the inequalityhas infinitely many solutions.

scholarly journals Diophantine approximation with mixed powers of primes

2018 ◽  
Vol 14 (07) ◽  
pp. 1903-1918
Author(s):  
Wenxu Ge ◽  
Huake Liu
Keyword(s):  
Real Number ◽  
Diophantine Approximation ◽  
Infinitely Many Solutions ◽  
Real Numbers

Let [Formula: see text] be an integer with [Formula: see text], and [Formula: see text] be any real number. Suppose that [Formula: see text] are nonzero real numbers, not all the same sign and [Formula: see text] is irrational. It is proved that the inequality [Formula: see text] has infinitely many solutions in primes [Formula: see text], where [Formula: see text], and [Formula: see text] for [Formula: see text]. This generalizes earlier results. As application, we get that the integer parts of [Formula: see text] are prime infinitely often for primes [Formula: see text].

Diophantine Approximation and Horocyclic Groups

1957 ◽  
Vol 9 ◽  
pp. 277-290 ◽  
Author(s):  
R. A. Rankin
Keyword(s):  
Real Number ◽  
Diophantine Approximation ◽  
Modular Group ◽  
Positive Real Number ◽  
Irrational Number ◽  
Infinitely Many Solutions ◽  
Positive Real ◽  
Closed Set ◽  
Coprime Integers ◽  
Image Position

1. Introduction. Let ω be an irrational number. It is well known that there exists a positive real number h such that the inequality(1)has infinitely many solutions in coprime integers a and c. A theorem of Hurwitz asserts that the set of all such numbers h is a closed set with supremum √5. Various proofs of these results are known, among them one by Ford (1), in which he makes use of properties of the modular group. This approach suggests the following generalization.

A NOTE ON INHOMOGENEOUS DIOPHANTINE APPROXIMATION IN BETA-DYNAMICAL SYSTEM

2014 ◽  
Vol 91 (1) ◽  
pp. 34-40 ◽  
Author(s):  
YUEHUA GE ◽  
FAN LÜ
Keyword(s):  
Dynamical System ◽  
Hausdorff Dimension ◽  
Real Number ◽  
Lebesgue Measure ◽  
Diophantine Approximation ◽  
Positive Function ◽  
Real Numbers ◽  
Image Position

AbstractWe study the distribution of the orbits of real numbers under the beta-transformation$T_{{\it\beta}}$for any${\it\beta}>1$. More precisely, for any real number${\it\beta}>1$and a positive function${\it\varphi}:\mathbb{N}\rightarrow \mathbb{R}^{+}$, we determine the Lebesgue measure and the Hausdorff dimension of the following set:$$\begin{eqnarray}E(T_{{\it\beta}},{\it\varphi})=\{(x,y)\in [0,1]\times [0,1]:|T_{{\it\beta}}^{n}x-y|<{\it\varphi}(n)\text{ for infinitely many }n\in \mathbb{N}\}.\end{eqnarray}$$

scholarly journals Diophantine approximation with one prime, two squares of primes and one kth power of a prime

2021 ◽  
Vol 19 (1) ◽  
pp. 373-387
Author(s):  
Alessandro Gambini
Keyword(s):  
Real Number ◽  
Diophantine Approximation ◽  
Infinitely Many Solutions ◽  
Real Numbers

Abstract Let 1 < k < 14 / 5 1\lt k\lt 14\hspace{-0.08em}\text{/}\hspace{-0.08em}5 , λ 1 , λ 2 , λ 3 {\lambda }_{1},{\lambda }_{2},{\lambda }_{3} and λ 4 {\lambda }_{4} be non-zero real numbers, not all of the same sign such that λ 1 / λ 2 {\lambda }_{1}\hspace{-0.08em}\text{/}\hspace{-0.08em}{\lambda }_{2} is irrational and let ω \omega be a real number. We prove that the inequality ∣ λ 1 p 1 + λ 2 p 2 2 + λ 3 p 3 2 + λ 4 p 4 k − ω ∣ ≤ ( max ( p 1 , p 2 2 , p 3 2 , p 4 k ) ) − ψ ( k ) + ε | {\lambda }_{1}{p}_{1}+{\lambda }_{2}{p}_{2}^{2}+{\lambda }_{3}{p}_{3}^{2}+{\lambda }_{4}{p}_{4}^{k}-\omega | \le {\left(\max \left({p}_{1},{p}_{2}^{2},{p}_{3}^{2},{p}_{4}^{k}))}^{-\psi \left(k)+\varepsilon } has infinitely many solutions in prime variables p 1 , p 2 , p 3 , p 4 {p}_{1},{p}_{2},{p}_{3},{p}_{4} for any ε > 0 \varepsilon \gt 0 , where ψ ( k ) = min 1 14 , 14 − 5 k 28 k \psi \left(k)=\min \left(\frac{1}{14},\frac{14-5k}{28k}\right) .

Diophantine approximation by unlike powers of primes

2017 ◽  
Vol 13 (09) ◽  
pp. 2445-2452 ◽  
Author(s):  
Zhixin Liu
Keyword(s):  
Real Number ◽  
Diophantine Approximation ◽  
Infinitely Many Solutions ◽  
Real Numbers

Let [Formula: see text] be nonzero real numbers not all of the same sign, satisfying that [Formula: see text] is irrational, and [Formula: see text] be a real number. In this paper, we prove that for any [Formula: see text] [Formula: see text] has infinitely many solutions in prime variables [Formula: see text].

scholarly journals A problem of Hooley in Diophantine approximation

1996 ◽  
Vol 38 (3) ◽  
pp. 299-308
Author(s):  
Glyn Harman
Keyword(s):  
Real Number ◽  
Diophantine Approximation ◽  
Infinitely Many Solutions ◽  
Image Position

In [5] Professor Hooley announced without proof the following result which is a variant of well-known work by Heilbronn [4]and Danicic [3] (see [1]).Let k≥2 be an integer, b a fixed non-zero integer, and a an irrational real number. Then, for any ɛ> 0, there are infinitely many solutions to the inequalityHere

Analogue of a theorem of Khintchine in fields of formal power series

1966 ◽  
Vol 62 (4) ◽  
pp. 637-642 ◽  
Author(s):  
T. W. Cusick
Keyword(s):  
Power Series ◽  
Positive Constant ◽  
Classical Result ◽  
Real Numbers ◽  
Linear Forms ◽  
Absolute Value ◽  
The Difference ◽  
Image Position ◽  
Formal Power ◽  
Positive Integers

For a real number λ, ‖λ‖ is the absolute value of the difference between λ and the nearest integer. Let X represent the m-tuple (x1, x2, … xm) and letbe any n linear forms in m variables, where the Θij are real numbers. The following is a classical result of Khintchine (1):For all pairs of positive integers m, n there is a positive constant Г(m, n) with the property that for any forms Lj(X) there exist real numbers α1, α2, …, αn such thatfor all integers x1, x2, …, xm not all zero.

scholarly journals On Diophantine approximation by unlike powers of primes

2019 ◽  
Vol 17 (1) ◽  
pp. 544-555
Author(s):  
Wenxu Ge ◽  
Weiping Li ◽  
Tianze Wang
Keyword(s):  
Diophantine Approximation ◽  
Infinitely Many Solutions ◽  
Real Numbers

Abstract Suppose that λ1, λ2, λ3, λ4, λ5 are nonzero real numbers, not all of the same sign, λ1/λ2 is irrational, λ2/λ4 and λ3/λ5 are rational. Let η real, and ε > 0. Then there are infinitely many solutions in primes pj to the inequality $\begin{array}{} \displaystyle |\lambda_1p_1+\lambda_2p_2^2+\lambda_3p_3^3+\lambda_4p_4^4+\lambda_5p_5^5+\eta| \lt (\max{p_j^j})^{-1/32+\varepsilon} \end{array}$. This improves an earlier result under extra conditions of λj.

Approximation by a sum of polynomials of different degrees involving primes

1979 ◽  
Vol 27 (4) ◽  
pp. 454-466
Author(s):  
Ming-Chit Liu
Keyword(s):  
Infinitely Many Solutions ◽  
Real Numbers ◽  
Integer Coefficients ◽  
Image Position

AbstractLet λj (1 ≦ j ≦ 4) be any nonzero real numbers which are not all of the same sign and not all in rational ratio and let pj be polynomials of degree one or two with integer coefficients and positive leading coefficients. The author proves that if exactly two pj are of degree two then for any real n there are infinitely many solutions in primes pj of the inequality . where 0 <β < (√(21)–1)∖5760.

scholarly journals Uniform Diophantine approximation related to -ary and -expansions

2014 ◽  
Vol 36 (1) ◽  
pp. 1-22 ◽  
Author(s):  
YANN BUGEAUD ◽  
LINGMIN LIAO
Keyword(s):  
Hausdorff Dimension ◽  
Real Number ◽  
Diophantine Approximation ◽  
Large Integer ◽  
Real Numbers

Let $b\geq 2$ be an integer and $\hat{v}$ a real number. Among other results, we compute the Hausdorff dimension of the set of real numbers ${\it\xi}$ with the property that, for every sufficiently large integer $N$, there exists an integer $n$ such that $1\leq n\leq N$ and the distance between $b^{n}{\it\xi}$ and its nearest integer is at most equal to $b^{-\hat{v}N}$. We further solve the same question when replacing $b^{n}{\it\xi}$ by $T_{{\it\beta}}^{n}{\it\xi}$, where $T_{{\it\beta}}$ denotes the classical ${\it\beta}$-transformation.

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