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PreprintCC BY 4.0arXiv

Combinatorics of translations of meager and closed measure zero sets

  • Aleksander Cieślak
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Abstract

We study combinatorial properties of translations of meager and closed sets of measure zero. We discuss constellations of Borel conjectures for related classes of small sets and show that there are no uncountable null-additive sets in the Miller model. We also study cardinal invariants of those classes. In particular, we investigate the cardinal invariants of σ-ideals ℋF related to meager-additive sets. We obtain a new characterization of additivity of meager ideal and answer some questions of Cardona. We also show that non(ℰ*), that is equal to the translation version of the covering number of the ideal ℰ, is close to the cardinal invariant related to the Laver property. This strengthens a result of Bartoszyński and Judah and a result of Elekes and Steprāns. Finally, we show that every ℰ-Luzin set is in ℰ*.

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Cite

BibTeXRIS
BibTeX
@misc{cieslak2026combinatorics,
  title = {{Combinatorics of translations of meager and closed measure zero sets}},
  author = {Aleksander Cieślak},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.28669},
  eprint = {2609.28669},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.28669},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Combinatorics of translations of meager and closed measure zero sets
AU  - Aleksander Cieślak
PY  - 2026
DA  - 2026/09/23
PB  - arXiv
DO  - 10.48550/arxiv.2609.28669
UR  - https://arxiv.org/abs/2609.28669
AB  - We study combinatorial properties of translations of meager and closed sets of measure zero. We discuss constellations of Borel conjectures for related classes of small sets and show that there are no uncountable null-additive sets in the Miller model. We also study cardinal invariants of those classes. In particular, we investigate the cardinal invariants of σ-ideals ℋ_F related to meager-additive sets. We obtain a new characterization of additivity of meager ideal and answer some questions of Cardona. We also show that non(ℰ^*), that is equal to the translation version of the covering number of the ideal ℰ, is close to the cardinal invariant related to the Laver property. This strengthens a result of Bartoszyński and Judah and a result of Elekes and Steprāns. Finally, we show that every ℰ-Luzin set is in ℰ^*.
ER  -