Mathematics Colloquium
Date: Wednesday, July 13th, 2022
Time: 3:30 pm to 4:30 pm
Venue: Lecture Hall 6
On some interesting sequences in diophantine approximation
Veekesh Kumar
NISER, Bhubaneshwar.
13-07-22

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Abstract

For a real number x, let || x || denote the distance from x to the nearest integer. The study of the sequence || \alpha^n || for \alpha > 1 naturally arises in various contexts in number theory. For example, it is not known that the sequence || e^n || tends to zero as n tends to infinity. Also, the growth of the sequence || (3/2)^n || is linked to the famous Waring's problem. In 1919, Hardy proved that if \alpha > 1 is an algebraic number such that || \alpha^n || tends to 0, then \alpha is an algebraic integer. Later, Pisot showed that if \alpha>1 is such that \sum_{n=1}^{\infty} || \alpha^n ||^2 converges, then \alpha is an algebraic integer. In this talk, I will give an overview of these developments and discuss my recent result about the growth of the sequence || \lambda \alpha^n +\beta ||. This can be viewed as an inhomogeneous analogue of Roth's inequality with moving targets.