Let $c$ be an irrational real number. Let $\{\cdot\}$ be the fractional part operator. I would like to get some sense of how in-the-dark we are about the distribution of values of $\{cn!\}$, for familiar values of $c$. This is related to a previous post which (essentially) asks the question "Does $n!/(2\pi)$ tend to a limit mod 1?"

Here is the question: Can anyone give a value of $c$ which is either algebraic, or a familiar transcendental, or defined in some reasonably simple way using the elementary functions of calculus, such that

- $\{cn!\}<1/2$ infinitely often, or
- $\{cn!\}$ tends to a limit, or
- The values of $\{cn!\}$ are dense in the interval $[0,1]$.

What do I know that we know about all this? First of all, it is a theorem of P. Diaconis (The Annals of Probability 1977, v5) that $\log(n!)$ is uniformly distibuted mod 1. This has the consequence that any sequence of leading (most significant) digits appears infinitely often. This is probably not going to be of any direct help, but it seems like it deserves to be mentioned.

Secondly, and importantly, it is known that for any lacunary sequence of positive integers $a_n$ (meaning that there is a fixed $\rho>1$ such that the inequality $a_{n+1}>\rho a_n$ holds for all large enough $n$) there are real numbers $c$ such that the sequence $\{ca_n\}$ is bounded away from 0 mod 1, and in fact the set of such real numbers has Hausdorff dimension 1. This is trivial to prove for $\rho > 2$, and in fact in this case we can easily choose $c$ (nonconstuctively!) to get any of the behaviors described in Items 1,2 and 3.

For $\rho$ near 1 the above statement about lacunary sequences was an Erdos problem, first solved by B. de Mathan (Acta Math. Hungar. 1980 v36). There is an exposition by Katznelson here.

Since the sequence $n!$ is lacunary (with $\rho$ as large as one wants) we already know that the behaviors described in Items 1-3 occur in abundance. The question is whether we know any specific examples.