Haar-$\mathcal I$ sets: looking at small sets in Polish groups through compact glasses

Abstract

Generalizing Christensen's notion of a Haar-null set and Darji's notion of a Haar-meager set, we introduce and study the notion of a Haar-I set in a Polish group. Here I is an ideal of subsets of some compact metrizable space K. A Borel subset B ⊂ X of a Polish group X is called Haar-I if there exists a continuous map f : K → X such that f −1 (B + x) ∈ I for all x ∈ X. Moreover, B is generically Haar-I if the set of witness functions {f ∈ C(K, X) : ∀x ∈ X f −1 (B + x) ∈ I} is comeager in the function space C(K, X). We study (generically) Haar-I sets in Polish groups for many concrete and abstract ideals I, and construct the corresponding distinguishing examples. We prove some results on Borel hull of Haar-I sets, generalizing results of Solecki, Elekes, Vidnyánszky, Doležal, Vlasǎk on Borel hulls of Haar-null and Haar-meager sets. Also we establish various Steinhaus properties of the families of (generically) Haar-I sets in Polish groups for various ideals I.

A Haar meager set that is not strongly Haar meager

“…We remark that in the case of Haar null sets [1,Theorem 4.3] answers the analogous question affirmatively.…”

“…For the compact metric space K 0 and function f 0 defined in Definition 4.8, if g ∈ Z ω , then f −1 0 (R + g) is a nowhere dense subset of K 0 . (Using the terminology of [1], this states that R is Haar nowhere dense.) Proof.…”

“…We will use recursion to define a sequence x ∈ K + t(K) ⊆ Z ω . First let (1) x n = u n. After this we will select the elements of the sequence x one by one. Assume that we already defined x j for an index j ≥ n and…”

“…Remark. In the previous claim the implication (2) ⇒ (1) remains true for any abelian Polish group G. This can be proved by slightly modifying the proof of [1,Theorem 5.13].…”

“…The result [1,Theorem 5.13] shows that in a certain class of Polish groups the Haar meager sets and the strongly Haar meager sets coincide:…”