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. This was the motivation for Mahler in 1957 to prove that for any non-integral rational number $\alpha > 1$ and any real number $c$ with $0 < c < 1$, the inequality $\|\alpha^n\| < c^n$ has only finitely many solutions in $n\in\mathbb{N}$. Mahler also asked the characterization of all algebraic numbers satisfying the same property as the non-integral rational numbers. In 2004, Corvaja and Zannier proved a Thue-Roth-type inequality with moving targets and as consequence, they completely answered the above question of Mahler. In this talk, we will explore this theme and will present recent result, building on the earlier works of Corvaja and Zannier, establishing an inhomogeneous Thue-Roth’s type theorem with moving targets.

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Last updated: 02 Nov 2024