Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a 1periodic function. I am looking about the conditions on $(a,b)\in\mathbb{R}^2$ such that we have the property : $$\frac{1}{n}\sum_{\ell=0}^{n1}f(a+b\ell)\underset{n\rightarrow +\infty}{\longrightarrow}\int_0^1 f(x)dx.$$ Thank you for your help !
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This follows from Weyl's equidistribution theorem. When $b$ is irrational, convergence holds whenever $f$ is continuous, or even just Riemannintegrable (see http://individual.utoronto.ca/hannigandaley/equidistribution.pdf for an exposition). Integrability is not enough by itself, since you can make the sum zero by changing $f$ on the countable set $\{a+b\ell\}$, which has measure zero. When $b$ is rational, convergence certainly doesn't hold for most $f$, since $a+b\ell$ takes on just finitely many values modulo $1$. 

