Infinite combinatorics and the foundations of regular variation

Indagationes Mathematicae

2013

Self Cite

Section: The Foundational Questionmentioning

confidence: 99%

Section: The Foundational Questionmentioning

confidence: 99%

See 1 more Smart Citation

The Steinhaus theorem and regular variation: de Bruijn and after

Indagationes Mathematicae

2013

Self Cite

“…Foremost among these are the Ash-Erdős-Rubel paper [AER], where a growth condition is used instead, and the work of Heiberg [Hei] and Seneta [Sen1], [Sen2], where side-conditions involving the limsup are imposed instead. Informed by the viewpoint of [BOst1], we generalize the results of these papers.…”

mentioning

confidence: 92%

“…The outstanding foundational question of the theory-raised and left open in [BG1], [BG2], [BGT]-is what common generalization of measurability and the Baire property suffices. This question is answered in [BOst1], where we obtain sets of conditions on h, each necessary and sufficient for UCT (see Theorem UCT below). In [BOst2] this motivates a unified approach to the Karamata theory of the two cases by regarding each as a subfamily of a single family of functions, one that is defined by combinatorial properties shared by both.…”

mentioning

confidence: 99%

Very slowly varying functions. II

2009

Colloq. Math.

Abstract. This paper is a sequel to papers by Ash, Erdős and Rubel, on very slowly varying functions, and by Bingham and Ostaszewski, on foundations of regular variation. We show that generalizations of the Ash-Erdős-Rubel approach-imposing growth restrictions on the function h, rather than regularity conditions such as measurability or the Baire property-lead naturally to the main result of regular variation, the Uniform Convergence Theorem.1. Introduction. We work with the Karamata theory of regular and slow variation; see [BGT] for a monograph account. Here the main result is the Uniform Convergence Theorem (UCT) which asserts that the defining pointwise convergence for slow variation in fact holds uniformly on compact sets if the function h in question is either (Lebesgue) , where we obtain sets of conditions on h, each necessary and sufficient for UCT (see Theorem UCT below). In [BOst2] this motivates a unified approach to the Karamata theory of the two cases by regarding each as a subfamily of a single family of functions, one that is defined by combinatorial properties shared by both. An alternative unification (see [BOst3]) derives the measure and category forms of their shared infinite combinatorics from a single topological result, the Category Embedding Theorem, by specialization to two topologies-the Euclidean topology and the density topology.A very few papers in regular variation are able to make progress without imposing regularity conditions. Foremost among these are the Ash-Erdős-Rubel paper [AER], where a growth condition is used instead, and the work of Heiberg [Hei]

Automatic continuity: subadditivity, convexity, uniformity

2009

Aequat. Math.

We examine various related instances of automatic properties of functions -that is, cases where a weaker property necessarily implies a stronger one under suitable side-conditions, e.g. connecting geometric and combinatorial features of their domains. The side-conditions offer a common approach to (mid-point) convex, subadditive and regularly varying functions (the latter by way of the uniform convergence theorem). We consider generic properties of the domain sets in the side-conditions -properties that hold typically, or off a small exceptional set. The genericity aspects develop earlier work of Kestelman [Kes] and of Borwein and Ditor [BoDi]. The paper includes proofs of three new analytic automaticity theorems announced in [BOst7].