Selection Principles in Mathematics — Problem 43 ([BG])

Definitions and mathematical statement

For the formulation below, use the standard interpretation of ${\mathbb{Z}}^{{\mathbb{N}}}$ as the additive product of discrete copies of ${\mathbb{Z}}$, with ${\mathbb{N}}=\{1,2,\ldots\}$, and give every subgroup the subspace topology. These topology and metric conventions are implicit rather than explicit in the supplied source. Equip ${\mathbb{Z}}^{{\mathbb{N}}}$ with the usual product metric \[ d(x,y)=\begin{cases}0,&x=y,\\2^{-\min\{n\in{\mathbb{N}}:x(n)\ne y(n)\}},&x\ne y.\end{cases} \] Here strong measure zero is interpreted with respect to the restriction of this metric to the subgroup.

For $f,h\in{\mathbb{N}}^{{\mathbb{N}}}$, or for nonnegative integer-valued sequences, write $f\leq^*h$ if $f(n)\leq h(n)$ for all but finitely many $n$. For $g\in{\mathbb{Z}}^{{\mathbb{N}}}$, put $|g|(n)=|g(n)|$. A subgroup $G\leq{\mathbb{Z}}^{{\mathbb{N}}}$ is bounded if \[ \exists b\in{\mathbb{N}}^{{\mathbb{N}}}\ \forall g\in G\quad |g|\leq^*b. \] Thus $G$ is unbounded precisely when \[ \forall b\in{\mathbb{N}}^{{\mathbb{N}}}\ \exists g\in G\ \forall N\in{\mathbb{N}}\ \exists n\geq N\quad |g(n)|>b(n). \]

The metric space $G$ has strong measure zero if, for every sequence $(\varepsilon_n)_{n\in{\mathbb{N}}}$ of positive real numbers, there are sets $A_n\subseteq G$ such that \[ G=\bigcup_{n\in{\mathbb{N}}}A_n, \qquad \operatorname{diam}_d(A_n)<\varepsilon_n\quad(n\in{\mathbb{N}}), \] where the diameter of the empty set is taken to be zero.

Let $\mathcal O(G)$ be the family of open covers of $G$. The selection property $\mathsf S_{\mathrm{fin}}(\mathcal O,\mathcal O)$ means that, for every sequence $(\mathcal U_n)_{n\in{\mathbb{N}}}$ from $\mathcal O(G)$, there are finite subfamilies $\mathcal F_n\subseteq\mathcal U_n$ with \[ G=\bigcup_{n\in{\mathbb{N}}}\bigcup\mathcal F_n. \] Its failure therefore means that there exists a sequence of open covers $(\mathcal U_n)_{n\in{\mathbb{N}}}$ such that \[ \forall (\mathcal F_n)_{n\in{\mathbb{N}}}\in\prod_{n\in{\mathbb{N}}}[\mathcal U_n]^{<\omega},\qquad G\ne\bigcup_{n\in{\mathbb{N}}}\bigcup\mathcal F_n. \]

Consistency question. Is it consistent that there exists a subgroup $G\leq{\mathbb{Z}}^{{\mathbb{N}}}$ which has strong measure zero, is unbounded with respect to $\leq^*$ as defined above, and fails $\mathsf S_{\mathrm{fin}}(\mathcal O,\mathcal O)$? Under the usual interpretation of consistency relative to $\mathsf{ZFC}$, the question asks whether \[ \operatorname{Con}(\mathsf{ZFC})\ \Longrightarrow\ \operatorname{Con}\!\left(\mathsf{ZFC}+\text{there exists such a subgroup }G\right). \] The source does not explicitly specify the underlying set theory; this is an interpretation requiring review.

Supplied context. The source states, for subgroups of ${\mathbb{Z}}^{{\mathbb{N}}}$, \[ \begin{aligned} G\models\mathsf S_1(\Omega_{\mathrm{nbd}},\Gamma) &\quad\Longleftrightarrow\quad G\text{ is bounded} &&\text{[Bab05]},\\ G\models\mathsf S_1(\mathcal O_{\mathrm{nbd}},\mathcal O_{\mathrm{nbd}}) &\quad\Longleftrightarrow\quad G\text{ has strong measure zero} &&\text{[coc11]}. \end{aligned} \] These are retained as supplied background assertions, without independent verification. Here $\mathsf S_1(\mathcal A,\mathcal B)$ denotes selecting one member from each family in a sequence from $\mathcal A$ so that the selected members form an element of $\mathcal B$. The precise conventions for the neighborhood-cover families and $\Gamma$ are not supplied in the record and are not needed for the direct formulation above.

Short English statement

The question asks whether it is consistent that an additive subgroup of ${\mathbb{Z}}^{{\mathbb{N}}}$ has strong measure zero, has no single eventual bound on the absolute values of all its elements, and fails the Menger covering property $\mathsf S_{\mathrm{fin}}(\mathcal O,\mathcal O)$. The formulation uses the usual product topology and product metric; these conventions and the interpretation of consistency relative to $\mathsf{ZFC}$ require confirmation from the source. The registry labels the problem open; that label is supplied curation, not an independently verified status assessment.

Sources

• Local UnsolvedMath registry, permanent ID 800615, exact source problem code catalogue entry, supplied complete source record. The registry records proposer Tsaban, proposed year 2006, and status open.
• Tsaban, Selection Principles and Special Sets of Reals: Open Problems (2006), Problem 43, as identified in the supplied provenance: https://arxiv.org/abs/math/0609601. The supplied record reports access on 2026-10-16 and extraction by arxiv_latex; this access report is not an independent verification of the source.
• The source title and background attach the attribution/context label [BG]. Its expansion is not supplied.
• The background cites [Bab05] for the boundedness equivalence and [coc11] for the strong-measure-zero equivalence. Full bibliographic entries and links for these labels are not supplied.