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. 130722 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.
