Products of K-Analytic Sets in Locally Compact Groups and Kuczma–Ger Classes

We prove that for any K-analytic subsets A,B of a locally compact group X if the product AB has empty interior (and is meager) in X, then one of the sets A or B can be covered by countably many closed nowhere dense subsets (of Haar measure zero) in X. This implies that a K-analytic subset A of X can be covered by countably many closed Haar-null sets if the set AAAA has an empty interior in X. It also implies that every non-open K-analytic subgroup of a locally compact group X can be covered by countably many closed Haar-null sets in X (for analytic subgroups of the real line this fact was proved by Laczkovich in 1998). Applying this result to the Kuczma–Ger classes, we prove that an additive function f:X→R on a locally compact topological group X is continuous if and only if f is upper bounded on some K-analytic subset A⊆X that cannot be covered by countably many closed Haar-null sets.