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.

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 KuipersNiederreiter, 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 KuipersNiederreiter: 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.) 


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. 

