On thin-complete ideals of subsets of groups

Given a family $F$ of subsets of a group $G$ we describe the structure of its thin-completion $\tau^(F)$, which is the smallest thin-complete family that contains $I$. A family $F$ of subsets of $G$ is called thin-complete if each $F$-thin subset of $G$ belongs to $F$. A subset $A$ of $G$ is called $F$-thin if for any distinct points $x,y$ of $G$ the intersection $xA\cap yA$ belongs to the family $F$. We prove that the thin-completion of an ideal in an ideal. If $G$ is a countable non-torsion group, then the thin-completion $\tau^(F_G)$ of the ideal $F_G$ of finite subsets of $G$ is coanalytic but not Borel in the power-set $P_G$ of $G$.

💡 Research Summary

The paper introduces and studies the notion of thin‑completeness for families of subsets of a group (G). For a given family (F\subseteq\mathcal P(G)) a set (A\subseteq G) is called (F)-thin if for any two distinct group elements (x\neq y) the intersection (xA\cap yA) belongs to (F). This generalises the classical thin‑set concept (where the intersection is required to be finite) by allowing an arbitrary reference family (F).

A family (F) is thin‑complete if every (F)-thin set already lies in (F). For any initial family (F) there exists a smallest thin‑complete family containing it; this is the thin‑completion (\tau^{}(F)). The authors construct (\tau^{}(F)) via a transfinite iteration of the operator (\tau):
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