Infinite combinatorics in function spaces: Category methods

Abstract

The in…nite combinatorics here give statements in which, from some sequence, an in…nite subsequence will satisfy some conditionfor example, belong to some speci…ed set. Our results give such statements generically -that is, for 'nearly all'points, or as we shall say, for quasi all points -all o¤ a null set in the measure case, or all o¤ a meagre set in the category case. The prototypical result here goes back to Kestelman in 1947 and to Borwein and Ditor in the measure case, and can be extended to the category case also. Our main result is what we call the Category Embedding Theorem (CET), which contains the Kestelman-Borwein-Ditor Theorem (KBD) as a special case. Our main contribution is to obtain functionwise rather than pointwise versions of such results. We thus subsume results in a number of recent and related areas, concerning e.g. additive, subadditive, convex and regularly varying functions.

Beurling slow and regular variation

“…Then h n converges to the identity in the supremum metric, so (wcc) holds by [13,Theorem 6.2] (First Verification Theorem), and so Theorem CET above applies for the Euclidean case; applicability in the measure case is established as [11,Corollary 4.1]. (This is the basis on which the affine group preserves negligibility.)…”

“…The increasing sequence of sets {H x k (η)} covers S. So, for some k, the set H x k (η) is non-negligible. As H x k (η) is non-negligible, by (11), for some l the set…”

The Steinhaus theorem and regular variation: de Bruijn and after

“…The KestelmanBorwein-Ditor theorem is of this type. It turns out that one can often work 'generically', obtaining the desired property 'quasi-everywhere'-everywhere o¤ a 'negligible'set (meagre in the Baire case, null in the measure case); see [BinO2] for results of this type in a function-space setting. The main result of [BinO2] is there called the Category Embedding Theorem, a tool used in our subsequent papers.…”

Beyond Lebesgue and Baire II: Bitopology and measure-category duality

“…Applications of Th. KBD are wide ranging: in addition to the UCT of Sections 3 and 4 they include automatic continuity ([BOst6], [BOst7], [BOst-SteinOstr]), the theory of subadditive functions [BOst5], combinatorics in function spaces [BOst9] and more generally in topological groups and normed groups [BOst12]. For an extension see [BOst10].…”