Diophantine approximation of complex numbers

Normal numbers and Diophantine approximation

We begin by recalling some classical results on normal and nonnormal numbers. Then, we discuss the following general question. Take a property of Diophantine approximation (e.g., to be badly approximable by rational numbers, to be a Liouville number, etc.) and a property concerning the digits (e.g., to be normal, to lie in the middle third Cantor set, etc.), do there exist real numbers having b...

Diophantine approximation by conjugate algebraic numbers

In 1969, Davenport and Schmidt provided upper bounds for the approximation of a real number by algebraic integers. Their novel approach was based on the geometry of numbers and involved the duality for convex bodies. In the present thesis we study the approximation of a real number by conjugate algebraic numbers. We find inspiration in Davenport and Schmidt’s method, but ultimately our approxim...

Diophantine approximation and Diophantine equations

The first course is devoted to the basic setup of Diophantine approximation: we start with rational approximation to a single real number. Firstly, positive results tell us that a real number x has “good” rational approximation p/q, where “good” is when one compares |x − p/q| and q. We discuss Dirichlet’s result in 1842 (see [6] Course N◦2 §2.1) and the Markoff–Lagrange spectrum ([6] Course N◦1...

Complex Dimensions of Self-Similar Fractal Strings and Diophantine Approximation

Approximation of Complex Numbers by Cyclotomic Integers

We present a new method of approximating complex numbers by cyclotomic integers in Ze 2i=2 n ] whose coeecients with respect to the basis given by powers of e 2i=2 n are bounded in absolute value by a given integer M. It has been suggested by Cozzens and Finkelstein 5] that such approximations reduce the dynamic range requirements of the discrete Fourier transform. For xed n our algorithm gives...