On translations of subsets of the real line

Abstract

Abstract. In this paper we discuss various questions connected with translations of subsets of the real line. Most of these questions originate from W. Sierpiński. We discuss the number of translations a single subset of the reals may have. Later we discuss almost invariant subsets of Abelian groups.

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

“…An immediate consequence of Theorem 3.5 is the fact that Borel perfectly small subsets of 2 N generate an invariant σ-ideal on 2 N . This generalizes a result of Cichoń, Jasiński, Kamburelis and Szczepaniak (see [4,Proposition 4.4]) stating that if R is the union of two uncountable Borel sets A and B, then…”

“…Proof. The idea of the proof is closely related to a reasoning described at the end of [4,Section 4] by Cichoń, Jasiński, Kamburelis and Szczepaniak.…”

Set-Theoretic Properties of Schmidt's Ideal

“…Proof. A theorem of Rothberger (see for instance Theorem 7.3 in [5]) states that if J 0 and J 1 are proper, uniform, translation invariant, and orthogonal σ-ideals on R, then ℵ 1 ≤ cov(J 0 ) ≤ non(J 1 ). We therefore infer that cov(S) ≤ non(K) and non(S) ≥ cov(K).…”

On Partitions of the Real Line into Continuum Many Thick Subsets

“…Actually, this measure-theoretical property completely characterizes Bernstein sets in R (see e.g. [3], [8]).…”