Bugeaud, Y.; Dodson, M. M.; Kristensen, S. Zero-infinity laws in diophantine approximation. (English) Zbl 1204.11117 Q. J. Math. 56, No. 3, 311-320 (2005). Summary: It is shown that for any translation invariant outer measure \(M\), the \(M\)-measure of the intersection of any subset of \(\mathbb R^n\) that is invariant under rational translations and which does not have full Lebesgue measure with the closure of an open set of positive measure cannot be positive and finite. Analogues for \(p\)-adic fields and fields of formal power series over a finite field are established. The results are applied to some problems in metric Diophantine approximation. Cited in 5 Documents MSC: 11J83 Metric theory Keywords:translation invariant outer measure; metric Diophantine approximation; Zero-infinity laws PDFBibTeX XMLCite \textit{Y. Bugeaud} et al., Q. J. Math. 56, No. 3, 311--320 (2005; Zbl 1204.11117) Full Text: DOI arXiv