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We consider the question of how approximations satisfying Dirichlet's theoremspiral around vectors in $\mathbb{R}^d$. We give pointwise almost everywhereresults (using only the Birkhoff ergodic theorem on the space of lattices). Inaddition, we show that for $\textit{every}$ unimodular lattice, on average, thedirections of approximates spiral in a uniformly distributed fashion on the$d-1$ dimensional unit sphere. For this second result, we adapt a very recentproof of Marklof and Str\"ombergsson \cite{MS3} to show a spherical averageresult for Siegel transforms on$\operatorname{SL}_{d+1}(\mathbb{R})/\operatorname{SL}_{d+1}(\mathbb{Z})$. Ourtechniques are elementary. Results like this date back to the work ofEskin-Margulis-Mozes \cite{EMM} and Kleinbock-Margulis \cite{KM} and havewide-ranging applications. We also explicitly construct examples in which thedirections are not uniformly distributed.

Spiraling of approximations and spherical averages of Siegel transforms