(by $\{x\}$ I mean the fraction part of the real number $x$) If $a$ is an irrational number and $n$ is a integral number, what is the distribution of $\{na\}$? I'm asking for some continuous function $f:[0,1]\to\mathbb R$ such $\int_{\alpha}^{\beta}f(x)\;dx$ gives the probability that $\{na\}$ falls between $\alpha$ and $\beta$. When I calculated it for a bunch of irrational number, from $n=1$ to $10000$, I found that it's very close to uniform distribution. It's well known that {na} with a proper choose of $a$ could be arbitrary close to any real number in the $[0,1]$ interval. But this claim is more than that and wants the distribution to be uniform. I think that a quite simple simple proof may exist: If $a$ was rational, say $p/q$, a uniform discrete distribution have been existed. I mean if $n$ goes to infinity the number would fall into $[i/q,(i+1)/q]$ interval with probability $1/q$. Now If we could approximate $a$ with a rational $p/q$, with "sufficiently small" error, the same would happen for a. That is, {na}s would also fall into the $[i/q,(i+1)/q]$ with probability $1/q$. if $q$ goes to infinity the distribution would become continuous. And at last ... I think $a = p/q + c/(q^2)$ where $c$ is smaller than or equal to one, is a sufficiently good approximation. Good in the sense that such an approximation causes a uniform distribution.
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The distribution is known to be uniform (a result due to Weyl, I believe). An excellent reference for this (and much else) is Dym and McKean's book on harmonic analysis. 


For rational $a$ the answer (explicit bound for the error term) is given by Ostrowski's theorem (Ostrowski A. Bemerkungen zur Theorie der Diophantischen Approximationen,Abh. Math. Sem Hamburg, 1922, 1, s. 7798). It depends on the sum of partial quotients in continued fraction expansion of number $a$. For real number it is sufficient to take good rational approsimation (one of convegents). See also Khintchine A. Ya. Ein Satz uЁber KettenbruЁche, mit arithmetischen Anwendungen. — Mathematische Zeitschrift, 18: 1 (1923), 289–306. 

