We consider the question of how approximations satisfying Dirichlet's theorem spiral around vectors in $\mathbb{ R}^d$. We give pointwise almost everywhere results (using only the Birkhoff ergodic theorem on the space of lattices). In addition, we show that for $\textit{ every}$ unimodular lattice, on average, the directions of approximates spiral in a uniformly distributed fashion on the $d-1$ dimensional unit sphere. For this second result, we adapt a very recent proof of Marklof and Str\"ombergsson \cite{ MS3} to show a spherical average result for Siegel transforms on $\operatorname{ SL}_{ d+1}(\mathbb{ R})/\operatorname{ SL}_{ d+1}(\mathbb{ Z})$. Our techniques are elementary. Results like this date back to the work of Eskin-Margulis-Mozes \cite{ EMM} and Kleinbock-Margulis \cite{ KM} and have wide-ranging applications. We also explicitly construct examples in which the directions are not uniformly distributed.
Categories: math.NT math.DS