Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Borel Conjecture for the Marczewski ideal

Joerg Brendle, Wolfgang Wohofsky

TL;DR

The paper addresses a Marczewski-type Borel Conjecture (MBC) for the ideal $s_0$ of Marczewski null sets by analyzing translations in the Cantor group ${}^\omega 2$ and extending to arbitrary Polish groups. It proves a ZFC result: no set of size $\mathfrak c$ is $s_0$-shiftable, using dense, translation-invariant families of skew Sacks trees and transitive variants of s0-nullness; it also develops a Luzin/Sierpiński-type toolkit adapted to $s_0$ and leverages forcing arguments. In the Cohen model, MBC holds for larger continua, showing consistency of the conjecture beyond CH and giving a forcing framework via $G$-matrices and speed-functions to prevent new reals from lying in any $x+[T]$. The paper further generalizes the ZFC result to arbitrary Polish groups and provides a robust combinatorial/forcing toolkit (skew trees, transitive nullness, dense matrices) for studying smallness notions defined via translations, with potential implications for the structure of $s_0$-shiftable sets in diverse topological group settings.

Abstract

We show in ZFC that there is no set of reals of size continuum which can be translated away from every set in the Marczewski ideal. We also show that in the Cohen model, every set with this property is countable.

Borel Conjecture for the Marczewski ideal

TL;DR

The paper addresses a Marczewski-type Borel Conjecture (MBC) for the ideal of Marczewski null sets by analyzing translations in the Cantor group and extending to arbitrary Polish groups. It proves a ZFC result: no set of size is -shiftable, using dense, translation-invariant families of skew Sacks trees and transitive variants of s0-nullness; it also develops a Luzin/Sierpiński-type toolkit adapted to and leverages forcing arguments. In the Cohen model, MBC holds for larger continua, showing consistency of the conjecture beyond CH and giving a forcing framework via -matrices and speed-functions to prevent new reals from lying in any . The paper further generalizes the ZFC result to arbitrary Polish groups and provides a robust combinatorial/forcing toolkit (skew trees, transitive nullness, dense matrices) for studying smallness notions defined via translations, with potential implications for the structure of -shiftable sets in diverse topological group settings.

Abstract

We show in ZFC that there is no set of reals of size continuum which can be translated away from every set in the Marczewski ideal. We also show that in the Cohen model, every set with this property is countable.
Paper Structure (13 sections, 20 theorems, 19 equations)

This paper contains 13 sections, 20 theorems, 19 equations.

Table of Contents

  1. Introduction
  2. Basic definitions and historical remarks
  3. The theorems of the paper
  4. Preliminary facts about Marczewski null sets
  5. The ZFC result
  6. Reduction to the set $D$
  7. Transitive versions of being Marczewski null
  8. Luzin sets
  9. Skew trees
  10. The regular case
  11. The singular case
  12. Skew perfect sets in arbitrary Polish groups
  13. Marczewski Borel Conjecture in the Cohen model

Key Result

Theorem 2

(ZFC) Let $X \subseteq {}^\omega 2$ with $|X| = {\mathfrak c}$. Then $X$ is not $s_0$-shiftable, i.e., there is a $Y \in s_0$ such that $X + Y = {}^\omega 2$.

Theorems & Definitions (44)

  • Definition 1
  • Theorem 2
  • Corollary 3
  • Theorem 4
  • Proposition 6
  • Theorem 7
  • proof : Proof (Sketch)
  • Lemma 8
  • proof
  • Definition 9
  • ...and 34 more