scholarly journals Higher-rank Bohr sets and multiplicative diophantine approximation

2019 ◽  
Vol 155 (11) ◽  
pp. 2214-2233 ◽  
Author(s):  
Sam Chow ◽  
Niclas Technau
Keyword(s):  
Diophantine Approximation ◽  
Continued Fractions ◽  
Structural Theory ◽  
Real Numbers ◽  
Successive Minima ◽  
Arbitrary Rank ◽  
Littlewood Conjecture ◽  
Higher Rank ◽  
Almost All

Gallagher’s theorem is a sharpening and extension of the Littlewood conjecture that holds for almost all tuples of real numbers. We provide a fibre refinement, solving a problem posed by Beresnevich, Haynes and Velani in 2015. Hitherto, this was only known on the plane, as previous approaches relied heavily on the theory of continued fractions. Using reduced successive minima in lieu of continued fractions, we develop the structural theory of Bohr sets of arbitrary rank, in the context of diophantine approximation. In addition, we generalise the theory and result to the inhomogeneous setting. To deal with this inhomogeneity, we employ diophantine transference inequalities in lieu of the three distance theorem.

scholarly journals Continued fractions with low complexity: transcendence measures and quadratic approximation

2012 ◽  
Vol 148 (3) ◽  
pp. 718-750 ◽  
Author(s):  
Yann Bugeaud
Keyword(s):  
Lower Bound ◽  
Continued Fractions ◽  
Low Complexity ◽  
Quadratic Approximation ◽  
Real Numbers ◽  
Partial Quotients ◽  
Block Complexity

AbstractWe establish measures of non-quadraticity and transcendence measures for real numbers whose sequence of partial quotients has sublinear block complexity. The main new ingredient is an improvement of Liouville’s inequality giving a lower bound for the distance between two distinct quadratic real numbers. Furthermore, we discuss the gap between Mahler’s exponent w2 and Koksma’s exponent w*2.

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.

Quadratic approximation in ℚp

2014 ◽  
Vol 11 (01) ◽  
pp. 193-209 ◽  
Author(s):  
Yann Bugeaud ◽  
Tomislav Pejković
Keyword(s):  
Prime Number ◽  
Continued Fractions ◽  
Quadratic Approximation ◽  
Almost All

Let p be a prime number. Let w2 and [Formula: see text] denote the exponents of approximation defined by Mahler and Koksma, respectively, in their classifications of p-adic numbers. It is well-known that every p-adic number ξ satisfies [Formula: see text], with [Formula: see text] for almost all ξ. By means of Schneider's continued fractions, we give explicit examples of p-adic numbers ξ for which the function [Formula: see text] takes any prescribed value in the interval (0, 1].

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].

The Discrepancy of (nkx)k=1∞ With Respect to Certain Probability Measures

2020 ◽  
Vol 71 (2) ◽  
pp. 573-597
Author(s):  
Niclas Technau ◽  
Agamemnon Zafeiropoulos
Keyword(s):  
Probability Measure ◽  
Lacunary Sequence ◽  
Probability Measures ◽  
Littlewood Conjecture ◽  
Almost All

Abstract Let $(n_k)_{k=1}^{\infty }$ be a lacunary sequence of integers. We show that if $\mu$ is a probability measure on $[0,1)$ such that $|\widehat{\mu }(t)|\leq c|t|^{-\eta }$, then for $\mu$-almost all $x$, the discrepancy $D_N(n_kx)$ satisfies $$\begin{equation*}\frac{1}{4} \leq \limsup_{N\to\infty}\frac{N D_N(n_kx)}{\sqrt{N\log\log N}} \leq C\end{equation*}$$for some constant $C>0$. This proves a conjecture of Haynes, Jensen and Kristensen and allows an improvement on their previous result relevant to an inhomogeneous version of the Littlewood conjecture.

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.

scholarly journals A generalized family of multidimensional continued fractions: TRIP Maps

2014 ◽  
Vol 10 (08) ◽  
pp. 2151-2186 ◽  
Author(s):  
Krishna Dasaratha ◽  
Laure Flapan ◽  
Thomas Garrity ◽  
Chansoo Lee ◽  
Cornelia Mihaila ◽  
...  
Keyword(s):  
Number Field ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Real Numbers ◽  
Multidimensional Continued Fractions ◽  
The Family ◽  
Before And After ◽  
Multidimensional Continued Fraction

Most well-known multidimensional continued fractions, including the Mönkemeyer map and the triangle map, are generated by repeatedly subdividing triangles. This paper constructs a family of multidimensional continued fractions by permuting the vertices of these triangles before and after each subdivision. We obtain an even larger class of multidimensional continued fractions by composing the maps in the family. These include the algorithms of Brun, Parry-Daniels and Güting. We give criteria for when multidimensional continued fractions associate sequences to unique points, which allows us to determine when periodicity of the corresponding multidimensional continued fraction corresponds to pairs of real numbers being cubic irrationals in the same number field.

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