Thursday 18 Feb 2016: Badly approximable points in Twisted Diophantine Approximation
Paloma Bengoechea - York
I will talk about the approximation of an n-dimensional real vector y by the integer multiples qx of another fixed arbitrary real vector x. This problem can be viewed in terms of toral rotations if we identify the torus with the cube [0,1)^n and we think of qx as the position of the origin after q rotations by x. I will define the concept of badly approximable vector in this context and will discuss the "size" of the set of badly approximable vectors. The case n=1 and a particular case in higher dimension have been recently established with the works of Kim, Tseng and Bugeaud-Harrap-Kristensen-Velani. In a recent work with N. Moshchevitin, we establish the case n>1 in all its generality. With N. Stepanova we study this same problem on manifolds.