TY - JOUR
T1 - Borel sets without perfectly many overlapping translations, III
AU - Rosłanowski, Andrzej
AU - Shelah, Saharon
N1 - Publisher Copyright:
© 2025 Elsevier B.V.
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 - http://www.scopus.com/inward/record.url?scp=85219562583&partnerID=8YFLogxK
U2 - 10.1016/j.apal.2025.103565
DO - 10.1016/j.apal.2025.103565
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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 -