For a subset $A\subseteq\mathbb T$ of the unit circle $\mathbb T=\{z\in\mathbb C:|z|=1\}$, let $\tau_A$ be the smallest topology on the additive group of integers $\mathbb Z$ such that for every $z\in A$ the homomorphism $\chi_z:\mathbb Z\to\mathbb T$, $\chi_z:n\mapsto z^n$ in continuous.
The topology $\tau_{\mathbb T}$ is called the Bohr topology on $\mathbb Z$ and plays an important role in Harmonic Analysis.
It is well-known that the topological space $(\mathbb Z,\tau_{\mathbb T})$ contains no non-trivial convergent sequences. This fact motivates untroducing the following two small uncountable cardinals.
Let $\mathfrak s(\mathbb Z)$ be the smallest cardinality of a subset $A\subseteq\mathbb T$ such that the topological space $(\mathbb Z,\tau_A)$ contains no non-trivial convergent sequences.
Let $\mathfrak{s_-}(\mathbb Z)$ be the smallest cardinality of subset $A\subseteq \mathbb T$ for which there exists an infinite set $I\subseteq\mathbb Z$ such that the set $I-I$ contains no non-trivial convergent sequence in $(\mathbb Z,\tau_A)$.
It can be shown that $$\mathfrak s\le\mathfrak s_-(\mathbb Z)\le\mathfrak s(\mathbb Z),$$ where $\mathfrak s$ is the splitting number (equal to the smallest cardinality of a family $\mathcal S$ of infinite subsets of $\omega$ such that for any infinite set $I\subseteq\omega$ there exists a set $S\in\mathcal S$ such that the sets $I\cap S$ and $I\setminus S$ both are infinite).
Problem 1. Are the strict inequalities $\mathfrak s<\mathfrak s_-(\mathbb Z)$ and $\mathfrak s_-(\mathbb Z)<\mathfrak s(\mathbb Z)$ consistent?
Problem 2. What are the relations of the cardinals $\mathfrak s(\mathbb Z)$ and $\mathfrak s_-(\mathbb Z)$ to the cardinal $\mathfrak s(\mathbb R)$, studied by Will Brian and Alan Dow?
