Dichotomy and infinite combinatorics: the theorems of Steinhaus and Ostrowski

Abstract

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.

The Steinhaus theorem and regular variation: de Bruijn and after

“…A comprehensive examination of results of this kind was recently given in [BinO11], from the point of view of in…nite combinatorics; see also [BinO6]. In particular, the subgroup dichotomy is relevant to the area of Ramsey theory [BinO8].…”

“…3.2.5, p.141). The answer to this emerged in the recent study [BinO11]. See [BinO8] for details, of both category methods and links with in…nite combinatorics -including such classic results as van der Waerden's theorem.…”

“…This concerns preservation of the Baire property and measurability under the Souslin operation. It would take us too far a…eld to discuss this important matter more fully here; for details and references, we refer to [BinO10], [BinO11], [Ost3], [Ost4]. 5.…”

Beurling regular variation, Bloom dichotomy, and the Gołąb–Schinzel functional equation

“…I,§3,Th. 4]), if measurable (Fréchet), if bounded on a non-null measurable set (Ostrowski's Theorem, re…ning Darboux's result for intervals), or on a non-meagre Baire set (Mehdi [Meh]); see [Kucz], or the more recent account in [BinO5]. Such automatic continuity results are mirrored in Beck's 'algebraic ‡ows'in a metric space, which when bounded by a monotone function of the ‡ow's distance from some set K are continuous at points of K ([Bec, Th.…”

“…We summarize from [BinO7] the combinatorial framework needed here: Baire and measurable cases are handled together by working bi-topologically, using the Euclidean topology in the Baire case (the primary case) and the density topology in the measure case; see [BinO2], [BinO5], [BinO4]. We work in the a¢ ne group Af f acting on (R; +) using the notation n (t) = c n t + z n ; where c n !…”