Diophantine Approximation and Transcendence

Saradha Natarajan () and Ravindranathan Thangadurai
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Saradha Natarajan: University of Mumbai, DAE Centre for Excellence in Basic Sciences
Ravindranathan Thangadurai: Harish-Chandra Research Institute, Department of Mathematics

Chapter Chapter 5 in Pillars of Transcendental Number Theory, 2020, pp 61-85 from Springer

Abstract: Abstract Diophantine approximation deals with the solubility of inequalities in integers. Dirichlet obtained one of the first type of such result in 1842 based on pigeon-hole principle. He showed that when $$\alpha $$ is irrational, there exist infinitely many rationals $$p/q (q>0)$$ such that $$\left| \alpha -\frac{p}{q}\right| 0$$ such that $$\left| \alpha -\frac{p}{q}\right| >\frac{c(\alpha )}{q^\kappa }$$with $$\kappa =n.$$ This led him to construct first examples of transcendental numbers.

Date: 2020
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DOI: 10.1007/978-981-15-4155-1_5

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