Last week Lam Wai Kit posted an excellent introductory article of ergodic theory and its applications to number theory. As stated in the note, one consequence of Birhoff recurrence theorem is that for any non-constant polynomial , the fractional part of the sequence can be arbitrarily close to .
In fact, it is a special case of the following well-known result.
Theorem 1 Suppose is a non-constant real polynomial with irrational leading coefficient, then is uniformly distributed mod 1.
The theorem is a consequence of Weyl’s criterion on uniform distribution.
Proposition 2 (Weyl’s criterion) Let be a sequence of real numbers. Then is uniformly distributed mod 1 if and only if for any non-zero integer ,
Proof: (Sketch). By Weierstrass’ theorem, any step function with support in can be approximated by linearly combinations of with norm.
Weyl’s criterion leads us to the investigation of the following summation
Here is an interval, are integers. Here and what follows, the summation runs through integers. Let be the number of integers in . Throughout the rest of the note, and are polynomials of degree with irrational leading coefficient.
We first deal with special case being linear.
are intervals (can be empty). In fact
where for , .
Proof: By induction on , for ,
Let , the above is
Suppose the statement is true for , then by the Cauchy-Schwarz inequality
As above, the innermost summand is
This completes the proof.
Proof of Theorem 1. Let , then is linear with leading coefficient irrational. Let for some non-zero integer . Let . By the previous proposition and proposition 3
The theorem then follows from Weyl’s criterion.