Element-wise description of the $\mathcal I$-characterized subgroups of the circle
Raffaele Di Santo, Dikran Dikranjan, Anna Giordano Bruno, Hans Weber
TL;DR
This work provides a complete element-wise description of the subgroups $t_{oldsymbol{u}}^{ abla}(oldsymbol{T})$ of the circle group $oldsymbol{T}$ characterized by a sequence $oldsymbol{u}$ of integers when $u_nigm|u_{n+1}$ and under a translation-invariant free $P$-ideal $ abla$ on $oldsymbol{N}$. The authors formulate a precise necessary-and-sufficient criterion: for every $A otin abla$, the membership depends on a dichotomy into $b$-bounded and $b$-divergent behavior via the supports of $x$ and the auxiliary sequences $(c_n)$ and $(b_n)$, encoded in conditions $(a_x)$ and $(b_x)$ and the auxiliary property $( ext{U}_A)$. They compare and refine existing results (DG1, Ghosh) and provide corollaries for classical cases such as $ abla= ext{Fin}$ and $ abla= abla_eta$, while introducing the $ abla$-splitting property and DL-type hypotheses to extend sufficiency proofs. The paper also supplies sufficient non-membership criteria for $ar{x}$ and discusses an appendix with an alternate proof path that relaxes or strengthens certain ideal conditions. Overall, the results advance the descriptive set-theoretic and harmonic-analytic understanding of $ abla$-characterized subgroups in the circle by giving a detailed, verifiable, element-wise criterion.
Abstract
According to Cartan, given an ideal $\mathcal I$ of $\mathbb N$, a sequence $(x_n)_{n\in\mathbb N}$ in the circle group $\mathbb T$ is said to {\em $\mathcal I$-converge} to a point $x\in \mathbb T$ if $\{n\in \mathbb N: x_n \not \in U\}\in \mathcal I$ for every neighborhood $U$ of $x$ in $\mathbb T$. For a sequence $\mathbf u=(u_n)_{n\in\mathbb N}$ in $\mathbb Z$, let $$t_{\mathbf u}^\mathcal I(\mathbb T) :=\{x\in \mathbb T: u_nx \ \text{$\mathcal I$-converges to}\ 0 \}.$$ This set is a Borel (hence, Polishable) subgroup of $\mathbb T$ with many nice properties, largely studied in the case when $\mathcal I = \mathcal F in$ is the ideal of all finite subsets of $\mathbb N$ (so $\mathcal F in$-convergence coincides with the usual one) for its remarkable connection to topological algebra, descriptive set theory and harmonic analysis. We give a complete element-wise description of $t_{\mathbf u}^\mathcal I(\mathbb T)$ when $u_n\mid u_{n+1}$ for every $n\in\mathbb N$ and under suitable hypotheses on $\mathcal I$. In the special case when $\mathcal I =\mathcal F in$, we obtain an alternative proof of a simplified version of a known result.