# Beyond Erd\H{o}s-Kunen-Mauldin: Singular sets with shift-compactness properties

## TL;DR

This paper explores the properties of shift-compact sets in real analysis, extending classical theorems to broader contexts and constructing special sets with singular features under certain set-theoretic assumptions.

## Contribution

It generalizes the Kestelman-Borwein-Ditor Theorem to Baire groups and locally compact groups and constructs co-analytic shift-compact sets with singular properties under $V=L$.

## Key findings

- Shift-compactness extends to Baire groups and locally compact groups.
- Construction of co-analytic shift-compact sets concentrated on rationals.
- Effective proofs enable new generalizations and singular set constructions.

## 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.