Ergodic theory of simple continued fractions
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Abstract
This thesis combines two fields of mathematics: number theory and ergodictheory (as part of dynamical systems). We study a special representationof numbers throughout the thesis: the simple continued fraction. We furtherinvestigate how simple continued fractions play a central role in approximatingreal numbers by rational numbers in the theory of Diophantine approximation.Simple continued fractions are also connected to a special measure-preservingtransformation on [0, 1). Using ergodic theory results, we prove many propertiesof continued fractions.