Centre for Discrete and Applicable Mathematics

 CDAM Research Report, LSE-CDAM-2007-15

July 2007 (Revised January 2009)

Innite Combinatorics and the theorems of Steinhaus and Ostrowski

N. H. Bingham and A. J. Ostaszewski

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 theorem for additive functions.

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