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](http://www.researchgate.net/publication/11742669_Ergodic_theorems_along_sequences_and_Hardy_fields), 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?"



Is $\lfloor \log(n!)\rfloor \alpha$ equidistributed on the unit circle?
Yes. This follows from Theorem 3.2 of my [paper](http://www.math.uvic.ca/faculty/aquas/papers/paper18.pdf) with Michael Boshernitzan, Gregori Kolesnik and Máté Wierdl, 'Ergodic Averaging Sequences'. 


