Not uniform distribution of {nr} fractional parts

Question

Can a sequence like $\{n!\alpha\}$ or $\{(n!+1)\alpha\}$ (fractional parts), where $\alpha$ is irrational, be not uniformly distributed in $(0,1)$? I know about Weyl's result for polynomials $p(n)\cdot\alpha$. David

Answer 1

Yes. The standard proof that $e$ is irrational rests on proving that $\{n!e\}$ lies strictly between $0$ and $1$, and in fact the estimate we get shows that it converges to $0$ (since it's strictly less than $\frac1n$), which makes it highly non-equidistributed.

Consequently, $\{(n!+1)e\} = \{n!e + e\}$ will converge to $\{e\} \approx 0.71828$ since $n!e$ is so close to an integer, so again it will be non-equidistributed.