+91-674-249-4082

Submitted by dinesh on 29 July, 2021 - 15:50

Date/Time:

Thursday, August 5, 2021 - 11:30 to 12:30

Venue:

Online (Google Meet)

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^\vareps

$$

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.

**Google Meet Link: **meet.google.com/rpj-qpwn-ows

**School of Mathematical Sciences**

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Tel: +91-674-249-4081

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