Question

It is not difficult to show (even without Weyl criterion) that the sequence $\sqrt{n}$, $n=1,2,\ldots$ is equidistributed mod 1. However, I need a reference to this result. Can you help me? Thanks.

Answer 1

Fejer's theorem: If $w(t)$ is a function with continuous first and second derivatives whose signs are eventually constant, and if $t \cdot w'(t)$ goes to infinity for $t$ to infinity, and $w(t)/t$ goes to zero, then $(w(n): n=1,2,3,...)$ is uniformly distributed.

In particular, for any fixed $b>0$ and $\alpha$ between $0$ and $1$, $b\cdot n^\alpha$ is uniformly distributed.

Reference: Hlawka, The theory of uniform distribution, page 23. Certainly also in Kuipers-Niederreiter, which I do not have here at the moment. (EDIT: After a bit of prodding, Google helped me to find Theorem 2.5 on page 13 in Kuipers-Niederreiter: If the sequence of differences $b_n:=a_{n+1}-a_n$ converges monotonically to zero, and $n b_n$ diverges to infinity, then $a_n$ is u.d. mod 1.)

Answer 2

My favorite reference on this is G. Polya and G. Szego, Problems and Theorems in Analysis, vol. 1, second part, Chap IV, section 4, see for example problem 174.