From: tao@sonia.math.ucla.edu (Terence Tao)
Subject: Re: Uniform distribution of polynomial values
Date: 1 Jan 2001 06:37:08 GMT
Newsgroups: sci.math.research
Summary: Distributions mod 1 of values of a polynomial at integers

In article <i44ifnv9g8rg@forum.mathforum.com>,
Bob Griffin  <mathgroup22@hotmail.com> wrote:
>If a is irrational number then the sequence a,2a,3a,4a,... is
>uniformly distributed modulo 1, this follows easily from Weyl criteria
>for uniform distribution.
>But what about non linear polynomial in a ?
>if p(n) = (a_n)*x^n + (a_n-1)*x^n-1 + ... a_1*x is a polynomial with
>degree n >= 2 then a necessary condition for the values
>p(0),p(1),p(2)... to have uniform distribution mod 1 is that at least
>one of the coefficients a_n,a_n-1,... a_1 is irrational.
>Is this condition also sufficient ? 
>

Yes, this is another theorem of Weyl.  It follows easily from the 
uniform distribution of {na} and the Van der Corput difference Lemma.
see e.g. Kuipers, Niederreiter, "Uniform distribution of sequences",
John Wiley and Sons, 1975.

Terry