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.

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

Kronecker is about sets, Weyl is about sequences.

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.

I have a feeling I've written this out on this site before.

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.

  • $\begingroup$ Thanks. I will look at the other answer as well. $\endgroup$ – daniel Feb 4 '12 at 4:57

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