Friendly Measures, Homogeneous Flows and Singular Vectors

The theory of diophantine approximation studies how well x = (x1, . . . , xn) ∈ R can be approximated by (p1/q1, . . . , pn/qn) ∈ Q of a given ‘complexity’, where this complexity is usually measured by the quantity lcm(q1, . . . , qn). Thus one is interested in minimizing the difference, in a suitable sense, between qx and a vector p, where p ∈ Z and q ∈ N, with a given upper bound on q. Often one finds that certain approximation problems admit a solution for almost every x, while others admit a solution for almost no x; one is then interested in understanding whether the typical properties remain typical when additional restrictions are placed on x. As an example, consider the notion of a singular vector, introduced by A. Khintchine in the 1920s (see [Kh, Ca]). Say that x is singular if for any δ > 0 there is T0 such that for all T ≥ T0 one can find p ∈ Z and q ∈ N with