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

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