Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a 1-periodic function. I am looking about the conditions on $(a,b)\in\mathbb{R}^2$ such that we have the property : $$\frac{1}{n}\sum_{\ell=0}^{n-1}f(a+b\ell)\underset{n\rightarrow +\infty}{\longrightarrow}\int_0^1 f(x)dx.$$ Thank you for your help !
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$\begingroup$ It might depend on $f$ (and if $f$ is not integrable then the RHS doesn't even exist...). It is well known that the limit holds for every continuous $f$ iff $b$ is irrational; one very fruitful approach is via Weyl's equidistribution criterion. $\endgroup$– Noam D. ElkiesCommented Jun 6, 2013 at 14:17
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This follows from Weyl's equidistribution theorem. When $b$ is irrational, convergence holds whenever $f$ is continuous, or even just Riemann-integrable (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 $1$.
answered Jun 6, 2013 at 14:16
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$\begingroup$ Actually, rational $b$ is not so bad, if $f$ is continuous and you choose $a$ carefully. $\endgroup$– user25199Commented Jul 23, 2013 at 12:40