References for the result that $\sqrt{n}$ is equidistributed mod 1 - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T02:14:55Z http://mathoverflow.net/feeds/question/107195 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/107195/references-for-the-result-that-sqrtn-is-equidistributed-mod-1 References for the result that $\sqrt{n}$ is equidistributed mod 1 Richard Bonne 2012-09-14T16:40:14Z 2012-09-15T14:30:24Z <p>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.</p> http://mathoverflow.net/questions/107195/references-for-the-result-that-sqrtn-is-equidistributed-mod-1/107196#107196 Answer by Goldstern for References for the result that $\sqrt{n}$ is equidistributed mod 1 Goldstern 2012-09-14T16:51:13Z 2012-09-14T17:09:32Z <p>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. </p> <p>In particular, for any fixed $b>0$ and $\alpha$ between $0$ and $1$, $b\cdot n^\alpha$ is uniformly distributed. </p> <p>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.)</p> http://mathoverflow.net/questions/107195/references-for-the-result-that-sqrtn-is-equidistributed-mod-1/107257#107257 Answer by Alexandre Eremenko for References for the result that $\sqrt{n}$ is equidistributed mod 1 Alexandre Eremenko 2012-09-15T14:30:24Z 2012-09-15T14:30:24Z <p>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. </p>