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.
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$\begingroup$ A tangentially related paper: ams.org/mathscinet-getitem?mr=2060024 $\endgroup$– John PardonSep 14, 2012 at 17:04
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$\begingroup$ How do you do it without Weyl? $\endgroup$– Igor RivinSep 14, 2012 at 17:12
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1$\begingroup$ @Rivin: See here isibang.ac.in/~sury/weyl.pdf $\endgroup$– Richard BonneSep 14, 2012 at 17:27
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2 Answers
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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.)
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1$\begingroup$ Do the proofs of those theorems use the Weyl criterion? $\endgroup$ Sep 14, 2012 at 23:11
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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.
answered Sep 15, 2012 at 14:30