Beyond Erdős-Kunen-Mauldin: Shift-compactness properties and singular sets

Abstract

The Kestelman-Borwein-Ditor Theorem asserts that a non-negligible subset of R which is Baire (= has the Baire property, BP) or measurable is shift-compact: it contains some subsequence of any null sequence to within translation by an element of the set. Here effective proofs are recognized to yield (i) analogous category and Haar-measure metrizable generalizations for Baire groups and locally compact groups respectively, and (ii) permit under V=L construction of co-analytic shift-compact subsets of R with singular properties, e.g. being concentrated on Q, the rationals.