A Note on the Distributions of (log n)mod 1

For sequences sufficiently close to (a log n), with an arbitrary real constant a, this note describes the precise asymptotics of the associated empirical distributions modulo one, with respect to the Kantorovich metric as well as a discrepancy-style metric. In particular, the note demonstrates how these asymptotics depend on a in a delicate, discontinuous way. The results strengthen and complement known facts in the literature.