Badly Approximable Points on Manifolds By: Victor Beresnevich

Summary This paper is motivated by two problems in the theory of Diophantine approximation, namely, Davenport’s problem regarding badly approximable points on submanifolds of a Euclidean space and Schmidt’s problem regarding the intersections of the sets of weighted badly approximable points. The problems have been recently settled in dimension two but remain open in higher dimensions. In this paper we develop new techniques that allow us to tackle them in full generality. The techniques rest on lattice points counting and a powerful quantitative result of Bernik, Kleinbock and Margulis. The main theorem of this paper implies that any finite intersection of the sets of weighted badly approximable points on any analytic nondegenerate submanifold of $$\mathbb {R}^n$$ R n has full dimension. One of the consequences of this result is the existence of transcendental real numbers badly approximable by algebraic numbers of any bounded degree.

Source: | San Francisco

Diophantine Approximation by Conjugate Algebraic Numbers By: Guillaume Alain

Summary 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 approximations come from the theory of continued fractions. We get a general optimal result for which we offer two different proofs. We then extend two of Davenport and Schmidt’s important results to the context of an imaginary quadratic number field. Our method follows that of Michel Laurent who simplified Davenport and Schmidt’s argument in 2003. One of their original results is optimal and so is our extension.

Source: | San Francisco