scholarly journals Further Variations on the Six Exponentials Theorem.

2005 ◽  
Vol Volume 28 ◽  
Author(s):  
Michel Waldschmidt
Keyword(s):  
Present Note ◽  
Rational Numbers ◽  
Algebraic Numbers ◽  
Linearly Independent ◽  
International Audience ◽  
Rank 2

International audience According to the Six Exponentials Theorem, a $2\times 3$ matrix whose entries $\lambda_{ij}$ ($i=1,2$, $j=1,2,3$) are logarithms of algebraic numbers has rank $2$, as soon as the two rows as well as the three columns are linearly independent over the field $\BbbQ$ of rational numbers. The main result of the present note is that one at least of the three $2\times 2$ determinants, viz. $$ \lambda_{21}\lambda_{12}-\lambda_{11}\lambda_{22}, \quad \lambda_{22}\lambda_{13}-\lambda_{12}\lambda_{23}, \quad \lambda_{23}\lambda_{11}-\lambda_{13}\lambda_{21} $$ is transcendental.

scholarly journals Joint Universality of the Zeta-Functions of Cusp Forms

Mathematics ◽  
2021 ◽  
Vol 9 (17) ◽  
pp. 2161
Author(s):  
Renata Macaitienė
Keyword(s):  
Cusp Form ◽  
Modular Group ◽  
Zeta Functions ◽  
Rational Numbers ◽  
Cusp Forms ◽  
Algebraic Numbers ◽  
Joint Universality ◽  
Universality Theorem ◽  
Linearly Independent ◽  
Joint Universality Theorem

Let F be the normalized Hecke-eigen cusp form for the full modular group and ζ(s,F) be the corresponding zeta-function. In the paper, the joint universality theorem on the approximation of a collection of analytic functions by shifts (ζ(s+ih1τ,F),⋯,ζ(s+ihrτ,F)) is proved. Here, h1,⋯,hr are algebraic numbers linearly independent over the field of rational numbers.

An application of the Thue–Siegel–Roth theorem to elliptic functions

1971 ◽  
Vol 69 (1) ◽  
pp. 157-161 ◽  
Author(s):  
J. Coates
Keyword(s):  
Number Theory ◽  
Diophantine Equations ◽  
Elliptic Functions ◽  
Rational Numbers ◽  
Algebraic Numbers ◽  
Number Of Solutions ◽  
Transcendental Numbers ◽  
Linearly Independent ◽  
Image Position ◽  
Effective Proof

Let α1, …, αn be n ≥ 2 algebraic numbers such that log α1,…, log αn and 2πi are linearly independent over the field of rational numbers Q. It is well known (see (6), Ch. 1) that the Thue–Siegel–Roth theorem implies that, for each positive number δ, there are only finitely many integers b1,…, bn satisfyingwhere H denotes the maximum of the absolute values of b1, …, bn. However, such an argument cannot provide an explicit upper bound for the solutions of (1), because of the non-effective nature of the theorem of Thue–Siegel–Roth. An effective proof that (1) has only a finite number of solutions was given by Gelfond (6) in the case n = 2, and by Baker(1) for arbitrary n. The work of both these authors is based on arguments from the theory of transcendental numbers. Baker's effective proof of (1) has important applications to other problems in number theory; in particular, it provides an algorithm for solving a wide class of diophantine equations in two variables (2).

Productly linearly independent sequences

2015 ◽  
Vol 52 (3) ◽  
pp. 350-370
Author(s):  
Jaroslav Hančl ◽  
Katarína Korčeková ◽  
Lukáš Novotný
Keyword(s):  
Rational Numbers ◽  
Infinite Products ◽  
Linearly Independent ◽  
New Concepts ◽  
Infinite Sequences

We introduce the two new concepts, productly linearly independent sequences and productly irrational sequences. Then we prove a criterion for which certain infinite sequences of rational numbers are productly linearly independent. As a consequence we obtain a criterion for the irrationality of infinite products and a criterion for a sequence to be productly irrational.

scholarly journals Ubiquity and large intersections properties under digit frequencies constraints

2008 ◽  
Vol 145 (3) ◽  
pp. 527-548 ◽  
Author(s):  
JULIEN BARRAL ◽  
STÉPHANE SEURET
Keyword(s):  
Hausdorff Dimension ◽  
Rational Numbers ◽  
Intersection Property ◽  
Algebraic Numbers ◽  
Real Numbers ◽  
Countable Intersection ◽  
The Real ◽  
Large Intersection ◽  
Digit Frequencies

AbstractWe are interested in two properties of real numbers: the first one is the property of being well-approximated by some dense family of real numbers {xn}n≥1, such as rational numbers and more generally algebraic numbers, and the second one is the property of having given digit frequencies in some b-adic expansion.We combine these two ways of classifying the real numbers, in order to provide a finer classification. We exhibit sets S of points x which are approximated at a given rate by some of the {xn}n, those xn being selected according to their digit frequencies. We compute the Hausdorff dimension of any countable intersection of such sets S, and prove that these sets enjoy the so-called large intersection property.

scholarly journals Classification of skew translation generalized quadrangles, I

2015 ◽  
Vol Vol. 17 no. 1 (Combinatorics) ◽  
Author(s):  
Koen Thas
Keyword(s):  
Generalized Quadrangles ◽  
New Classification ◽  
Open Problems ◽  
International Audience ◽  
Rank 2

Combinatorics International audience We describe new classification results in the theory of generalized quadrangles (= Tits-buildings of rank 2 and type B2), more precisely in the (large) subtheory of skew translation generalized quadrangles (``STGQs''). Some of these involve, and solve, long-standing open problems.

The Equality of a Manifold's Rank and Dimension

1969 ◽  
Vol 12 (2) ◽  
pp. 183-184
Author(s):  
R. S. D. Thomas
Keyword(s):  
Vector Fields ◽  
Present Note ◽  
Linearly Independent ◽  
Commuting Vector Fields

T. J. Willmore has shown that if a differentiable manifold's rank (the maximum number of everywhere linearly independent commuting vector fields definable on it) equals the manifold's dimension, then the manifold is a torus of the appropriate dimension [1]. This theorem is proved more simply and without any differentiability hypothesis in the present note.

Algorithmes des fractions continues et de Jacobi-Perron

1996 ◽  
Vol 53 (2) ◽  
pp. 341-350 ◽  
Author(s):  
Brigitte Adam ◽  
Georges Rhin
Keyword(s):  
Continued Fraction ◽  
Rational Numbers ◽  
Algebraic Numbers

We give an algorithm by which one can compute, using only rational numbers, the continued fraction and more generally the Jacobi-Perron algorithm expansion of real algebraic numbers.

Chowla’s theorem over function fields

2018 ◽  
Vol 14 (06) ◽  
pp. 1689-1698
Author(s):  
Yoshinori Hamahata
Keyword(s):  
Dirichlet Series ◽  
Function Field ◽  
Function Fields ◽  
Rational Numbers ◽  
Periodic Coefficients ◽  
Linearly Independent ◽  
Parity Condition

Sarvadaman Chowla proved that if [Formula: see text] is an odd prime, then [Formula: see text] ([Formula: see text]) are linearly independent over the field of rational numbers. We establish an analog of this result over function fields. As an application, we prove an analog of the Baker–Birch–Wirsing theorem about the non-vanishing of Dirichlet series with periodic coefficients at [Formula: see text] in the function field setup with a parity condition.

Singular n-tuples and Hausdorff dimension

1977 ◽  
Vol 81 (3) ◽  
pp. 377-385 ◽  
Author(s):  
R. C. Baker
Keyword(s):  
Hausdorff Dimension ◽  
Euclidean Space ◽  
Measure Zero ◽  
Rational Numbers ◽  
Linearly Independent ◽  
Singular Numbers ◽  
Image Position ◽  
Almost All ◽  
Dimensional Lebesgue Measure

1. Introduction. Throughout the paper θ = (θ1, …, θn), φ = (φ1, …, φn), … denote points of Euclidean space Rn. We write Kn for the set of θ in Rn for which θl, …, θn, 1 are linearly independent over the rational numbers. We denote points of the set of integer n-tuples Zn by x, y, … We writeIf α is a real number, ∥α∥ denotes the distance from α to the nearest integer.Let θ ∈ Rn. By a theorem of Dirichlet ((2), chapter 1, theorem VI).for all X ≥ 1. We say that θ is singular ifSingular points form a set of n-dimensional Lebesgue measure zero. In fact, H. Davenport and W. M. Schmidt (3) showed thatfor almost all θ in Rn. Although there are no singular numbers in Kl ((2), p. 94) there are ‘highly singular’ n-tuples in Kn for n ≥ 2.

scholarly journals Explicit logarithmic formulas of special values of hypergeometric functions 3F2

2019 ◽  
Vol 22 (05) ◽  
pp. 1950040
Author(s):  
Masanori Asakura ◽  
Toshifumi Yabu
Keyword(s):  
Linear Combination ◽  
Hypergeometric Function ◽  
Hypergeometric Functions ◽  
Rational Numbers ◽  
Explicit Description ◽  
Generalized Hypergeometric Function ◽  
Algebraic Numbers ◽  
Nagoya Math ◽  
Special Values ◽  
Geometric Study

In [M. Asakura, N. Otsubo and T. Terasoma, An algebro-geometric study of special values of hypergeometric functions [Formula: see text], to appear in Nagoya Math. J.; https://doi.org/10.1017/nmj.2018.36 ], we proved that the value of [Formula: see text] of the generalized hypergeometric function is a [Formula: see text]-linear combination of log of algebraic numbers if rational numbers [Formula: see text] satisfy a certain condition. In this paper, we present a method to obtain an explicit description of it.

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