Singular vectors in manifolds and countable intersections

Orateur: Barak WEISS
Localisation: ,
Type: Online Seminar in Diophantine Approximation and Related Topics
Site: N/A
Salle: Zoom
Date de début: 15/11/2022 - 14:00
Date de fin: 15/11/2022 - 15:00

A vector x = (x_1, ..., x_d) in R^d is totally irrational if 1, x_1, ..., x_d are linearly independent over rationals, and singular if for any epsilon, for all large enough T, there are solutions p in Z^d and q in {1, ..., T} to the inequality

||qx - p || < epsilon T^{-1/d}


In previous work we showed that certain smooth manifolds of dimension at least two, and certain fractals, contain totally irrational singular vectors. The argument for proving this is a variation on an old argument employed by Khintchine and Jarník. We now adapt this argument to show that for certain families of maps f_i: R^d -> R^{n_i}, certain manifolds contain points x such that f_i(x) is a singular vector for all i. This countable intersection property is motivated by some problems in approximation of vectors by vectors with coefficient in a number field. Joint work with Dmitry Kleinbock, Nikolaus Moshchevitin and Jacqueline Warren.