On Borel sets belonging to every invariant ccc $\sigma $-ideal on $2^{\mathbb {N}}$

Abstract

Let I ccc be the σ-ideal of subsets of the Cantor group 2 N generated by Borel sets which belong to every translation-invariant σ-ideal on 2 N satisfying the countable chain condition (ccc). We prove that I ccc strongly violates ccc. This generalizes a theorem of Balcerzak-Ros lanowski-Shelah stating the same for the σ-ideal on 2 N generated by Borel sets B ⊆ 2 N which have perfectly many pairwise disjoint translates. We show that the last condition does not follow from B ∈ I ccc even if B is assumed to be compact. Various other conditions which for a Borel set B imply that B ∈ I ccc are also studied. As a consequence we prove in particular that:• If A n are Borel sets, n ∈ N, and 2 N = n A n , then there is n 0 such that every perfect set P ⊆ 2 N has a perfect subset Q, a translate of which is contained in A n 0 . • CH is equivalent to the statement that 2 N can be partitioned into ℵ 1 many disjoint translates of a closed set.

Haar-smallest sets

“…However, we provide our own proof, which can be considered more natural and will be needed later in this section. Notice also that for the group 2 ω a far more general fact has been proved by Zakrzewski in [19,Proposition 3.12]: in every invariant ccc σ-ideal on 2 ω there is a compact set which is not Haar-countable. Proof.…”

“…In the mentioned cases we have a nice characterization, which connects notions introduced at the beginning of this paper with the above considerations (by showing that H[2 ω ] ≤ω = HCtbl, H[2 ω ] <ω = HFin and H[2 ω ] ≤n = Hn). Equivalence of items (b) and (c) in the following proposition is implicit in [19,Lemma 3.1].…”

“…Let us point out a connection of Haar-countable sets with covering the whole space by less than continuum translates of its subset: less than continuum translates of a Haar-countable set cannot cover the whole space (cf. [19,Lemma 4.1]). This fact links Haar-countable sets with the works of Dobrowolski, Elekes, Marciszewski, Miller, Steprāns and Tóth (see [9], [10], [11] and [18]).…”

“…Haar-n =⇒ Haar-(n + 1) =⇒ Haar-finite =⇒ Haar-countable for any n ∈ ω \ {0}. By removing the requirement about Borel hulls from the above definition, we would obtain the notions of perfectly κ-small sets studied by Zakrzewski in [19] in the context of ccc σ-ideals. The particular case of Haar-1 sets has been investigated by Balcerzak in [1], where he introduced the so-called property (D) of an invariant σ-ideal I, which says that there is a σ-compact Haar-1 set not belonging to I.…”

Differentiability of continuous functions in terms of Haar-smallness

“…Moreover, it is known that neither Haar-finite sets nor Haar-n sets form an ideal (see [13,Corollary 5.2 and Theorem 6.1]). Zakrzewski considered Haar-small sets in [16] under the name of perfectly κ-small sets. A particular case of Haar-1 sets was investigated by Balcerzak in [3].…”