TITLE: Is $\lfloor \log(n!)\rfloor \alpha$ equidistributed on the unit circle?
QUESTION [12 upvotes]: In this question $\lfloor a\rfloor$ means the greatest integer not exceeding $a$. 
Using van der Corput's inequalities one is able to show that $\log(n!)\alpha$ is equidistributed on the unit circle ($\mathbb{R}\setminus\mathbb{Z}$) for all $\alpha \in \mathbb{R}^{+}$.   
I am however having trouble trying to show that $\lfloor\log(n!)\rfloor\alpha$ is equidistributed on the unit circle for $\alpha \in \mathbb{R}\setminus\mathbb{Q}$. The reason I believe this result to hold is due to the research here, showing that $\lfloor\log(n!)\rfloor$ is a good sequence for mean $L^{2}$ convergence. Any ideas of how one could try and prove this would be much appreciated.
I suppose the more general question is: "Are there any results or methods that help in proving equidistribution (or good sequences for mean $L^{2}$ convergence) for functions that are rounded down?"

REPLY [15 votes]: Yes. This follows from Theorem 3.2 of my paper with Michael Boshernitzan, Gregori Kolesnik and Máté Wierdl, 'Ergodic Averaging Sequences'.