Yes, That's what happened. I'm still trying to work out the last details. Even though Don didn't phrase it this way, I think that it's an application of the Borel-Cantelli lemma: If E[n] are a series of events in a probability space and sum_n Prob(E[n]) < infinity, then the probability that infinitely many of these events occur is 0. To apply it to our problem, fix a bound M, and let E[n] be the event that |sum_{j=0}^{n-1} e(2^j theta) | < M and |sum_{j=0}^n e(2^j theta) | < M. I think that possibly you can let M depend on n. There is some sort of counting argument you can give, after noticing that the probability of E[n], depends on only the first n+1 bits of the binary expansion of theta, up to a small fudge. Victor On Tue, Dec 16, 2008 at 1:19 PM, Andy Latto <andy.latto@pobox.com> wrote:

On Mon, Dec 15, 2008 at 3:58 PM, victor miller <victorsmiller@gmail.com> wrote:

...

The idea for (2) is a counting argument: suppose that for a given M there is a T such that

1/2^T #{ 0 <= k < 2^T : there exists epsilon in [0,1) such that |sum_{n=0}^{T-1} e(2^-n k)| < M } < 1

Did this get posted before you finished? It doesn't seem like a sketch of a proof, just a single definition.

-- Andy.Latto@pobox.com