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Dichotomy and infinite combinatorics: the theorems of Steinhaus and Ostrowski

Published online by Cambridge University Press:  08 October 2010

Department of Mathematics, Imperial College London, London SW7 2AZ. e-mail:
Department of Mathematics, London School of Economics, London WC2A 2AE. e-mail:


We define combinatorial principles which unify and extend the classical results of Steinhaus and Piccard on the existence of interior points in the distance set. Thus the measure and category versions are derived from one topological theorem on interior points applied to the usual topology and the density topology on the line. Likewise we unify the subgroup theorem by reference to a Ramsey property. A combinatorial form of Ostrowski's theorem (that a bounded additive function is linear) permits the deduction of both the measure and category automatic continuity theorems for additive functions.

Research Article
Copyright © Cambridge Philosophical Society 2010

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