Suppose $\alpha$ irrational is not a Liouville number. i.e. we cannot find an a sequence of rational approximations $p_n/q_n$ with
$|\frac{p_n}{q_n}-\alpha|<\frac{1}{q_n^n}$.
I am trying an exercise from Einsiedler and Ward's "Ergodic theory with a view towards number theory" which asks me to prove that for an smooth $f$ on $\mathbb{T}$ (the circle) we have for all $x\in\mathbb{T}$
$|\frac{1}{N}\sum_{n=0}^{N-1}f(x+n\alpha)-\int_0^1 f(x) dx|<\frac{S}{N}$. $\qquad(*)$
Here $S$ is a constant independent of $x$ but could depend on $f,\alpha$ and $dx$ denotes the usual Lebesgue measure. Proving the other way is easy - for a Liouville number we can show $(*)$ is false. The idea is to note that we have bad sampling etc. But I have no idea how to gt a hold on this. Intuitively its obvious that the closer we can approximate $\alpha$ by rationals the less chance $(*)$ has of being true...
A few hints:
First consider the characters, for which the estimate can obtained explicitly (and works in fact for any irrational number $\alpha$).
Then sum over all the characters to obtain the Fourier series of $f$, which converges uniformly because the function is smooth.
Finally, estimate the right-hand side after summing over all the characters using the Cauchy-Schwarz inequality.