nt.number theory - Wanted: A constructive version of a theorem of Furstenberg and Weiss

Dear RJS,

I think Tim Gowers is right - the problem seems too hard. Reasonably good bounds are known on (for example) the least $n geq 1$ for which $Vert n^2 sqrt{2} Vert leq epsilon$; one can find such an $n$ with $n leq epsilon^{-7/4 + o(1)}$. This is a result of Zaharescu [Zaharescu, A; Small values of $n^2alphapmod 1$. Invent. Math. 121 (1995), no. 2, 379--388.] Zaharescu in fact obtains this result for any $theta$ in place of $sqrt{2}$. From a cursory glance at the paper I see that he uses the continued fraction expansion for $theta$ and so it may be that one can slightly improve his bound in the particular case of $sqrt{2}$.

It is an old conjecture of Heilbronn that the right bound here should be $epsilon^{-1 + o(1)}$. I don't know off the top of my head whether any more precise conjectures have been made based on sensible heuristics either for this or for your original problem.

To get an explicit upper bound for your problem one can proceed quite straightforwardly using arguments due to Weyl. I don't think this is the right place to describe an argument in detail: there are several variants, and I first learnt this from a Tim Gowers course at Cambridge. See Theorem 3.10 of these notes:

http://www.math.cmu.edu/~af1p/Teaching/AdditiveCombinatorics/notes-acnt.pdf

If you really had to show that 627 is the answer to your specific problem, probably the best bet would be to inspect all the quadratics $n^2sqrt{2} + theta n + theta'$ for $theta,theta'$ in some rather dense finite subset of $[0,1]^2$ and show using a computer that each takes (mod 1) a value less than 0.009999 for some $n leq 627$. Painful!

The argument of Furstenberg and Weiss uses ergodic theory and so will not directly lead to an effective bound.

There are quite detailed conjectures about the fractional parts of $n^2sqrt{2}$ (and other similar sequences) due to Rudnick, Sarnak and Zaharescu, essentially encoding the fact that this sequence of fractional parts is expected to behave like a Poisson process. I don't think those conjectures are likely to be helpful in your context since, taken too literally, they would seem to suggest that there are arbitrarily long intervals without a number such that $Vert n^2 sqrt{2} Vert < 0.01$ - contrary to Furstenberg-Weiss.

Nonetheless let me point out a recent paper of Heath-Brown which is very interesting in connection with these matters:

http://arxiv.org/pdf/0904.0714v1.

One more point perhaps worthy of mention: sequences with bounded gaps are usually known as syndetic.