Borel sets without perfectly many overlapping translations, III

Andrzej Rosłanowski; Saharon Shelah

doi:10.1016/j.apal.2025.103565

Borel sets without perfectly many overlapping translations, III

Andrzej Rosłanowski^*
, Saharon Shelah

^*Corresponding author for this work

Einstein Institute of Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We expand the results of Rosłanowski and Shelah [11,10] to all perfect Abelian Polish groups (H,+). In particular, we show that if α<ω₁ and 4≤k<ω, then there is a ccc forcing notion adding a Σ₂⁰ set B⊆H which has ℵ_α many pairwise k–overlapping translations but not a perfect set of such translations. The technicalities of the forcing construction led us to investigations of the question when, in an Abelian group, X−X⊆Y−Y imply that a translation of X or −X is included in Y.

Original language	English
Article number	103565
Journal	Annals of Pure and Applied Logic
Volume	176
Issue number	6
DOIs	https://doi.org/10.1016/j.apal.2025.103565
State	Published - Jun 2025

Bibliographical note

Publisher Copyright:
© 2025 Elsevier B.V.

Keywords

Abelian Polish groups
Borel sets
Forcing
Splitting rank

Access to Document

10.1016/j.apal.2025.103565License: CC BY-NC-ND

Cite this

@article{8433aac8b0b04dbcb26ea088e8c17362,

title = "Borel sets without perfectly many overlapping translations, III",

abstract = "We expand the results of Ros{\l}anowski and Shelah [11,10] to all perfect Abelian Polish groups (H,+). In particular, we show that if α<ω1 and 4≤k<ω, then there is a ccc forcing notion adding a Σ20 set B⊆H which has ℵα many pairwise k–overlapping translations but not a perfect set of such translations. The technicalities of the forcing construction led us to investigations of the question when, in an Abelian group, X−X⊆Y−Y imply that a translation of X or −X is included in Y.",

keywords = "Abelian Polish groups, Borel sets, Forcing, Splitting rank",

author = "Andrzej Ros{\l}anowski and Saharon Shelah",

note = "Publisher Copyright: {\textcopyright} 2025 Elsevier B.V.",

year = "2025",

month = jun,

doi = "10.1016/j.apal.2025.103565",

language = "אנגלית",

volume = "176",

journal = "Annals of Pure and Applied Logic",

issn = "0168-0072",

publisher = "Elsevier B.V.",

number = "6",

}

TY - JOUR

T1 - Borel sets without perfectly many overlapping translations, III

AU - Rosłanowski, Andrzej

AU - Shelah, Saharon

PY - 2025/6

Y1 - 2025/6

N2 - We expand the results of Rosłanowski and Shelah [11,10] to all perfect Abelian Polish groups (H,+). In particular, we show that if α<ω1 and 4≤k<ω, then there is a ccc forcing notion adding a Σ20 set B⊆H which has ℵα many pairwise k–overlapping translations but not a perfect set of such translations. The technicalities of the forcing construction led us to investigations of the question when, in an Abelian group, X−X⊆Y−Y imply that a translation of X or −X is included in Y.

AB - We expand the results of Rosłanowski and Shelah [11,10] to all perfect Abelian Polish groups (H,+). In particular, we show that if α<ω1 and 4≤k<ω, then there is a ccc forcing notion adding a Σ20 set B⊆H which has ℵα many pairwise k–overlapping translations but not a perfect set of such translations. The technicalities of the forcing construction led us to investigations of the question when, in an Abelian group, X−X⊆Y−Y imply that a translation of X or −X is included in Y.

KW - Abelian Polish groups

KW - Borel sets

KW - Forcing

KW - Splitting rank

UR - https://www.scopus.com/pages/publications/85219562583

U2 - 10.1016/j.apal.2025.103565

DO - 10.1016/j.apal.2025.103565

M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???

AN - SCOPUS:85219562583

SN - 0168-0072

VL - 176

JO - Annals of Pure and Applied Logic

JF - Annals of Pure and Applied Logic

IS - 6

M1 - 103565

ER -

Borel sets without perfectly many overlapping translations, III

Abstract

Bibliographical note

Keywords

Access to Document

Other files and links

Fingerprint

Cite this