The title says the problem. What are the values of $\gamma$ that satisfy equidistribution? As far as I know, $\forall x\in (\mathbb{Q}\cup {ne|n\in \mathbb{Z}})$, the sequence is not equidistributed.
2026-03-25 12:55:09.1774443309
When is a sequence $\{n!\gamma\}_{n=1}^{\infty}$ equidistributed in [0,1)?
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Korobov, Concerning some questions of uniform distribution, Izvestiya Akad. Nauk SSSR. Ser. Mat. 14 (1950) 215–238, MR0037876 (12, 321a), according to the review by Lowell Schoenfeld, proves that $\alpha n!$ is uniformly distributed if $\alpha=\sum_1^{\infty}[k^{1+\lambda}]/k!$ and $0<\lambda<1$. Schoenfeld goes on to write,
...other specializations of these results are also given. While the uniform distribution of these functions for almost all values of $\alpha$ (in the sense of measure 0) has been known since Weyl's work [Math. Ann. 77, 313–352 (1916)], not a single value of $\alpha$ has been known for which these functions actually are uniformly distributed.
Korobov's paper was in Russian. I don't know whether it has been translated into other languages. Several later papers cite this paper. The review by Olaf Stackelberg of Salat, Zu einigen Fragen der Gleichverteilung (mod 1), Czechoslovak Math. J. 18 (93) 1968 476–488, MR0229586 (37 #5160), gives more details on Korobov's result.