The Hardy–Littlewood problem for regular and uniformly distributed number sequences

Let $H$ be the set of functions $f(x)$ defined in $(0, 1)$, $f(0+0)=f(1-0)=+\infty$, monotone in neighborhoods of singular points and such that the improper Riemann integral $\int\limits_0^1f(x)\,dx$ converges. Let $Q$ be an arbitrary set of sequences $(\{x_i\})_{i=1}^\infty$ uniformly distributed in the interval $[0, 1]$. We find the set of those pairs in $H\times Q$ for which the following equality is valid:
$$ \lim\limits_{n\to\infty}\frac{1}{n}\sum_{i=1}^n f(\{x_i\})=\int\limits_0^1f(x)\,dx. $$