Let $f$ be a function of period 1, Riemann integrable on [0,1]. Let $\xi_n$ be a sequence which is equidistributed in $[0,1)$.
(a) Is it true that $$\frac{1}{N}\sum\limits_{n=1}^N f(x+\xi_n)$$ converges to the constant $\int_0^1 f(y) dy$ for each $x\in \mathbb{R}$ as $N\to \infty$ ?
(b) If so, is the convergence uniform over all $x$?