Beyond Erdős-Kunen-Mauldin: Singular sets with shift-compactness properties

Abstract

The Kestelman-Borwein-Ditor Theorem asserts that a non-negligible subset of $\mathbb{R}$ 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. 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 $\mathbb{Q}$, the rationals.

It is with regret that we announce that the first author Harry I. Miller (1939-2018) died on 16 Dec 2018, whilst finishing this paper. His last communication ended with: `I only wish I was 10 years younger (make that 15) so I could contribute (and learn) more."