Exact uniform approximation and Dirichlet spectrum

Abstract: We consider the Dirichlet spectrum, with respect to maximum norm and simultaneous approximation. It is basically the analogue of the famous (multi-dimensional) Lagrange spectrum with respect to uniform approximation. By Dirichlet’s Theorem it is contained in [0,1]. The central new result is that it equals the entire interval [0,1] when the number of variables is two or more. We thereby get a new, constructive proof of a recent result by Beresnevich, Guan, Marnat, Ramirez and Velani that there are Dirichlet improvable vectors that are neither bad nor singular, in any dimension. We provide several generalizations, including metrical claims.

dynamical systemsnumber theory

Audience: general audience

New England Dynamics and Number Theory Seminar