The Steinhaus theorem and regular variation: de Bruijn and after

Abstract

The contributions of N. G. de Bruijn to regular variation, and recent developments in this …eld, are discussed. A new version of the Uniform Convergence Theorem is given.

Beurling slow and regular variation

“…for all |t| |v|K −1 and all x X. So, in particular, for x X and t := v/ϕ(x), since |t| |v|K −1 , substitution in (16) yields…”

Effros, Baire, Steinhaus and non-separability

“…This theorem has wide-ranging consequences, including Steinhaus'Sumset Theorem -see the survey article [Ost4], and the recent [BinO3].…”

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

“…The Beurling theory of slow variation, originating in Beurling's generalization (for which see [Pet] and [Moh] -cf. [BinO6]) of the Wiener Tauberian Theorem, studies functions f with…”