Beyond measure zero we have yet another measure-y notion of smallness: *strong measure zero.* A set $S\subseteq\mathbb{R}$ is strong measure zero if, for any $f:\mathbb{N}\rightarrow\mathbb{R}\_{>0}$, there is a sequence $U\_i$ of open sets with


* the diameter of $U\_i$ is $<f(i)$, and
* $S\subseteq\bigcup\_{i\in\mathbb{N}} U\_i$.


(Various intermediate notions of nullness [are also considered](https://arxiv.org/pdf/1512.04424.pdf).) At least for me, it wasn't immediately obvious that there are measure zero sets which aren't strongly measure zero, but in fact the Cantor set is provably not strong measure zero.


Strong measure zero-ness raises a natural cardinal characteristic-style question:


* What is the smallest size $\mathfrak{s}\_-$ of a non-strongly measure zero set? What is the supremum $\mathfrak{s}\_+$ of the cardinalities of the strong measure zero sets?


Clearly every countable set is strong measure zero, so $\mathfrak{s}\_-$ is uncountable. The converse, Borel's conjecture, is consistent with ZFC (this was proved by Laver) and implies $\mathfrak{s}\_-=\omega\_1$ and $\mathfrak{s}\_+=\omega$ *(this reveals that $\mathfrak{s}\_+$ is a bit weird; still, it seems a natural thing to consider, even if it's not a "kosher" CCC)*. Meanwhile, Sierpinski showed that CH implies that $\mathfrak{s}\_+=\mathfrak{s}\_-=2^{\aleph\_0}$.


My question is:



> 
> What is known about $\mathfrak{s}\_-$ and $\mathfrak{s}\_+$? In particular, where does $\mathfrak{s}\_-$ sit amongst other cardinal characteristics (say, those in [Cichon's diagram](https://en.wikipedia.org/wiki/Cicho%C5%84%27s_diagram))?
> 
> 
> 



The "strong" measure number
Chapter 8 of "Set Theory On the structure of the real line" by Bartoszynski and Judah compiles a lot of combinatorial results about strong measure zero sets. Theorem 8.1.14 gives an old result of Arnie Miller. I'll use your notation $\mathfrak{s}\_-$ when I state it:


Theorem (Miller)
For every cardinal $\kappa$, the following are equivalent:


1. $\kappa\leq\mathfrak{s}\_-$
2. For every bounded family $F\subseteq\vphantom{b}^\omega\omega$ of size $<\kappa$ there is a function $g\in\vphantom{b}^\omega\omega$ such that for each $f\in F$ there are infinitely many $n$ with $g(n)=f(n)$.


That chapter has a ton of useful information about the structure of strong measure zero sets. 


