# On a strengthening of strong measure zero

Recall that a set of $X$ of reals has strong measure zero (SMZ) if for every sequence $\{\epsilon_n:n<\omega\}$ of positive real numbers, there is a sequence $\{I_n:n<\omega\}$ of intervals such that $I_n$ has length at most $\epsilon_n$ and $X\subseteq \bigcup_{n<\omega}I_n$.

By a theorem of Galvin, Mycielski, and Solovay, a set of reals $X$ has SMZ if and only if $X+F\neq\mathbb{R}$ for every meager $F\subseteq\mathbb{R}$.

I am looking for information about the collection of sets $X\subseteq\mathbb{R}$ which satisfy the stronger property that $X+F$ is meager for every meager $F$.

Clearly such sets have SMZ. My specific question is:

(1) Does this family of sets admit a more "combinatorial" definition? SMZ sets can be characterized several different ways: For example, a set $X\subseteq 2^\omega$ has SMZ if and only if for every partition $\{I_n:n<\omega\}$ of $\omega$ into intervals, there is a $z\in 2^\omega$ such that $$\forall x\in X\quad\exists^\infty n\in\omega\quad x\upharpoonright I_n = z\upharpoonright I_n$$. Is anything similar known for the situation I ask about?

However, I would also be satisfied with an answer to the following more general question:

(2) Has this collection of sets been systematically studied before? What are some references where one might learn more?

These questions are related to the field of selection principles. For example, Galvin and Miller have shown that $\gamma$-sets satisfy this stronger version of SMZ, and my graduate student Frank Ballone noted that their proof shows that sets of reals satisfying the selection principle $\binom{\Omega}{\mathcal{O}_\infty}$ (left undefined here, but due to Tsaban) do as well.

## 1 Answer

The sets you defined are called meager-additive, and the family of these sets is sometimes denoted $\mathcal{M}^\star$.

Google suggests a good number of references, and I recommend you also look at Barotszynski and Judah's set theory book. It must contain a combinatorial characterization of the kind you seek.

BTW, the result of Ballone that you cite is awesome!

• Many thanks, Boaz! A name is a powerful thing, as I spent a good part of the day trying to reason out what these sets must be called. Jun 13, 2016 at 22:01
• And Theorem 2.7.17 of Bartoszynski-Judah is exactly the sort of characterization I was looking for. I had pulled their book of the shelf to search earlier today, but I had been looking in Chapter 8 (where SMZ is dealt with in detail) rather than near the beginning... Jun 13, 2016 at 22:22
• Glad this helped. It would be nice if you could post here some link to Ballone's and your research on the topic, when available. Jun 14, 2016 at 12:12