scholarly journals Effective irrationality measures for quotients of logarithms of rational numbers

2015 ◽  
Vol Volume 38 ◽  
Author(s):  
Yann Bugeaud
Keyword(s):  
Rational Numbers ◽  
Linear Forms In Logarithms ◽  
Linear Forms ◽  
International Audience

International audience We establish uniform irrationality measures for the quotients of the logarithms of two rational numbers which are very close to 1. Our proof is based on a refinement in the theory of linear forms in logarithms which goes back to a paper of Shorey.

scholarly journals Linear forms in logarithms and exponential Diophantine equations

2020 ◽  
Vol Volume 42 - Special... ◽  
Author(s):  
Rob Tijdeman
Keyword(s):  
Diophantine Equations ◽  
Linear Forms In Logarithms ◽  
Linear Forms ◽  
Exponential Diophantine Equations ◽  
International Audience

International audience This paper aims to show two things. Firstly the importance of Alan Baker's work on linear forms in logarithms for the development of the theory of exponential Diophantine equations. Secondly how this theory is the culmination of a series of greater and smaller discoveries.

scholarly journals Quelques remarques sur des questions d'approximation diophantienne

1999 ◽  
Vol 59 (2) ◽  
pp. 323-334 ◽  
Author(s):  
Patrice Philippon
Keyword(s):  
Lower Bounds ◽  
Approximation Theory ◽  
Diophantine Approximation ◽  
Rational Numbers ◽  
Product Theorem ◽  
Linear Forms In Logarithms ◽  
Linear Forms ◽  
Subspace Theorem

Hoping for a hand-shake between methods from diophantine approximation theory and transcendance theory, we show how zeros estimates from transcendance theory imply Roth's type lemmas (including the product theorem). We also formulate some strong conjectures on lower bounds for linear forms in logarithms of rational numbers with rational coefficients, inspired by the subspace theorem and which would imply, for example, the abc conjecture.

scholarly journals Effective irrationality measures for real and p-adic roots of rational numbers close to 1, with an application to parametric families of Thue–Mahler equations

2016 ◽  
Vol 164 (1) ◽  
pp. 99-108
Author(s):  
YANN BUGEAUD
Keyword(s):  
Rational Numbers ◽  
Linear Forms In Logarithms ◽  
Special Situation ◽  
Large Power ◽  
Linear Forms ◽  
Parametric Families

AbstractWe show how the theory of linear forms in two logarithms allows one to get very good effective irrationality measures for nth roots of rational numbers a/b, when a is very close to b. We give a p-adic analogue of this result under the assumption that a is p-adically very close to b, that is, that a large power of p divides a−b. As an application, we solve completely certain families of Thue–Mahler equations. Our results illustrate, admittedly in a very special situation, the strength of the known estimates for linear forms in logarithms.

scholarly journals On the equation $a(x^m-1)/(x-1)=b(y^n-1)/(y-1)$: II

1984 ◽  
Vol Volume 7 ◽  
Author(s):  
Tarlok Nath Shorey
Keyword(s):  
Linear Forms In Logarithms ◽  
Linear Forms ◽  
International Audience

International audience Using the theory of linear forms in logarithms we generalize an earlier result with R. Balasubramanian on the equation of the title.

NOTE ON FOURIER–STIELTJES COEFFICIENTS OF COIN-TOSSING MEASURES

2020 ◽  
Vol 102 (3) ◽  
pp. 479-489
Author(s):  
XIANG GAO ◽  
SHENGYOU WEN
Keyword(s):  
Number Theory ◽  
Lower Bound ◽  
Transcendental Number ◽  
Linear Forms In Logarithms ◽  
Multiplicative Semigroup ◽  
Linear Forms ◽  
Coin Tossing

It is known that the Fourier–Stieltjes coefficients of a nonatomic coin-tossing measure may not vanish at infinity. However, we show that they could vanish at infinity along some integer subsequences, including the sequence ${\{b^{n}\}}_{n\geq 1}$ where $b$ is multiplicatively independent of 2 and the sequence given by the multiplicative semigroup generated by 3 and 5. The proof is based on elementary combinatorics and lower-bound estimates for linear forms in logarithms from transcendental number theory.

Applications of Linear Forms in Logarithms

2008 ◽  
pp. 411-440
Author(s):  
Yann Bugeaud ◽  
Guillaume Hanrot ◽  
Maurice Mignotte
Keyword(s):  
Linear Forms In Logarithms ◽  
Linear Forms
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