Some applications of Diophantine approximation. (English) Zbl 1045.11022

Bennett, M. A. (ed.) et al., Number theory for the millennium III. Proceedings of the millennial conference on number theory, Urbana-Champaign, IL, USA, May 21–26, 2000. Natick, MA: A K Peters (ISBN 1-56881-152-7/hbk). 261-284 (2002).
In this survey, the author gives some recent and not so recent applications of some of the most important Diophantine approximation techniques, i.e., linear forms in logarithms estimates and Thue-Siegel-Roth-Schmidt theory. Among other things, he mentions some results on gaps between integers composed of a given finite set of primes, transcendence results on infinite sums \(\sum_{n=1}^{\infty} f(n)/g(n)\) where \(f,g\) are polynomials with integer coefficients, transcendence results and non-vanishing results on series \(\sum_{n=1}^{\infty} f(n)/g(n)\) where \(f\) is a periodic completely multiplicative function and \(g\) is a polynomial with integer coefficients, and results on Diophantine equations such as \(ax^n-by^n=c\), Catalan’s equation, the Fermat-Catalan equation \(x^k+y^l=z^m\), Goormaghtigh’s equation \({x^m-1\over x-1}={y^n-1\over y-1}\), etc.
For the entire collection see [Zbl 1002.00007].


11D61 Exponential Diophantine equations
11J81 Transcendence (general theory)
11J68 Approximation to algebraic numbers
11J86 Linear forms in logarithms; Baker’s method