Geometric and arithmetic aspects of approximation vectors

Abstract:

Let x in R^d be a vector and let (p k, g k) in Z^d \times N denote its sequence of best approximationvectors, with respect to some norm. in the case d=1, the properties ofthis sequence for a.e. x are understood via the continued fraction algorithm, and the ergodic theory of this algorithm can be useoto obtain various limit laws such as the generic growth rate of the denominators, the distribution othe approximation, and more. In joint work with Uri Shapira, we extend many results in the one-dimensional setting, to d>1, and also study certain quantities associated with best approximationsthat have no one-dimensional analogues. The main technique, inspired by work of Cheung and Chevalier, is the use of a certain well-adapted cross-section to a diagonal flow on the space of lattices