Spiraling of approximations and spherical averages of Siegel transforms

We consider the question of how approximations satisfying Dirichlet's theorem spiral around vectors in Rd. We give pointwise almost everywhere results (using only the Birkhoff ergodic theorem on the space of lattices). In addition, we show that for 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 Strömbergsson [‘Free path lengths in quasicrystals’, Comm. Math. Phys. 330 (2014) 723–755] to show a spherical average result for Siegel transforms on SLd+1(R)/SLd+1(Z). Our techniques are elementary. Results like this date back to the work of Eskin–Margulis–Mozes [‘Upper bounds and asymptotics in a quantitative version of the Oppenheim conjecture’, Ann. of Math. 147 (1998) 93–141] and Kleinbock–Margulis [‘Bounded orbits of non-quasiunipotent flows on homogeneous spaces’, Amer. Math. Soc. Transl. 171 (1996) 141–172] and have wide-ranging applications. We also explicitly construct examples in which the directions are not uniformly distributed.