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Thursday, August 5, 2021 - 11:30 to 12:30
Online (Google Meet)
Veekesh Kumar
IMSc, Chennai
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.


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