From Kuiper's and Niederreiter's book Uniform distribution of sequences, Ch.2, § 3, I learn that an irrational number $\alpha\in \mathbf{R}\smallsetminus \mathbf{Q}$ is of constant type $\eta$ if the following supremum is finite: $$ \eta = \{\gamma>0 : \liminf_{q\to \infty} q^\gamma \langle q \alpha\rangle=0\}, $$ where $q$ ranges over integers and $$ \langle x\rangle = \inf_{z\in \mathbf{Z}} |x-z| . $$ The Thue–Siegel–Roth Theorem tells us that if $\alpha$ is algebraic, then $\alpha$ has constant type $1$.
Is there a reasonable description of the set of real numbers with constant type at least $2$?
