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Quasi-Monte Carlo rules are equal weight quadrature rules defined over thedomain $[0,1]^s$. Here we introduce quasi-Monte Carlo type rules for numericalintegration of functions defined on $\mathbb{R}^s$. These rules are obtained byway of some transformation of digital nets such that locally one obtains qMCrules, but at the same time, globally one also has the required distribution.We prove that these rules are optimal for numerical integration in fractionalBesov type spaces. The analysis is based on certain tilings of the Walsh phaseplane.

Quasi-Monte Carlo numerical integration on $\mathbb{R}^s$: digital nets and worst-case error