# Coming Events

## Seminar

Date/Time:
Thursday, August 5, 2021 - 11:30 to 12:30
Venue:
Speaker:
Veekesh Kumar
Affiliation:
IMSc, Chennai
Title:
On inhomogeneous extension of Thue-Roth's type inequality with moving targets

Let $\Gamma\subset \overline{\mathbb Q}^{\times}$ be a
finitely generated multiplicative group of algebraic numbers.  Let
$\delta, \beta\in\overline{\mathbb Q}^\times$  be  algebraic numbers
with $\beta$ irrational.  In this talk,  I will prove  that  there
exist only finitely many triples $(u, q, p)\in\Gamma\times\mathbb{Z}^2$
with $d = [\mathbb{Q}(u):\mathbb{Q}]$  such that
$$0<|\delta qu+\beta-p|<\frac{1}{H^\varepsilon(u)q^{d+\varepsilon}},$$
where $H(u)$ denotes  the absolute Weil height.  As an application of
this result, we also prove a transcendence result, which states as
follows:  Let $\alpha>1$ be a real number. Let $\beta$ be an algebraic
irrational and  $\lambda$ be a non-zero real algebraic number.   For a
given real number $\varepsilon >0$, if there are infinitely many
natural numbers $n$ for which  $||\lambda\alpha^n+\beta|| < 2^{- \varepsilon n}$ holds true, then  $\alpha$ is transcendental, where
$||x||$ denotes the distance from its nearest integer.