# Relationship between Kronecker's Approximation Thm and Weyl's Equidistribution Thm?

According to Prof. Wikipedia, the Equidistribution Theorem was proved by Weyl in 1910 and independently by two others around the same time.

The theorem states that for $\alpha$ irrational, the sequence $\alpha, 2\alpha, 3\alpha,..., etc. \pmod 1$ is uniformly distributed on the unit interval.

At Wolfram's MathWorld, a similar theorem is attributed to Kronecker, namely that for any irrational number $\alpha$, the sequence $\{n \alpha \}$ is dense in the unit interval, i.e., one can find a $\{k \alpha\}$ arbitrarily close to any real number b with $0 \leq b \leq 1$. This is styled the Kronecker Approximation Theorem. Kronecker died in 1891. Note: $\{n\alpha\}$ is the fractional part of $n\alpha$.

I haven't seen a proof of Kronecker's theorem, but I have seen a proof of Weyl's Theorem. Naively (as usual), it appears the two ideas can be used interchangeably in some contexts. Even more naively, it seems there is a strong theoretical relationship between the two.

Can someone offer a thumbnail sketch of the salient differences/similarities of the two ideas (or a good reference for same)? Knowing no better, I would say Weyl owed a considerable debt to Kronecker.

Thank you.

• Weyl's theorem is much stronger and implies Kronecker's theorem, which is much easier. – Qiaochu Yuan Feb 3 '12 at 21:59

If the sequence $u_1,u_2,\dots$ of elements of $[0,1)$ is uniformly distributed, then the underlying set $\lbrace\,u_1,u_2,\dots\,\rbrace$ is dense in $[0,1)$.
If a countable subset $S$ of $[0,1)$ is dense in $[0,1)$, then (by a theorem of Erdos) it can be ordered so as to be a uniformly distributed sequence - but it can also be ordered so as not to be uniformly distributed.
EDIT: found it: When is a sequence $(x_n) \subset [0,1]$ dense in $[0,1]$? Not exactly the same as the current question, but there is some overlap.