Selection Principles in Mathematics — Problem 55 ([prods])

Definitions and mathematical statement

For $d\in\{1,2\}$, equip ${\mathbb{R}}^d$ with its Euclidean topology, coordinatewise addition, and $d$-dimensional Lebesgue measure. A set is meager if it is a countable union of nowhere dense sets. A set is null if its Lebesgue outer measure is zero. For $A,B\subseteq{\mathbb{R}}^d$, put \[ A+B=\{a+b:a\in A,\ b\in B\}. \] A set $A\subseteq{\mathbb{R}}^d$ is meager-additive if $A+M$ is meager for every meager set $M\subseteq{\mathbb{R}}^d$. It is null-additive if $A+N$ is null for every null set $N\subseteq{\mathbb{R}}^d$. These definitions impose no additional measurability, definability, or cardinality assumptions on $A$.

The two questions supplied as Problem 55 are:

1. Is it true that, for every $X,Y\subseteq{\mathbb{R}}$, if both $X$ and $Y$ are meager-additive in ${\mathbb{R}}$, then $X\times Y$ is meager-additive in ${\mathbb{R}}^2$? Explicitly, must \[ \forall M\subseteq{\mathbb{R}}^2,\qquad M\text{ meager}\ \Longrightarrow\ (X\times Y)+M\text{ meager} \] hold under these hypotheses?
2. Is it true that, for every $X,Y\subseteq{\mathbb{R}}$, if both $X$ and $Y$ are null-additive in ${\mathbb{R}}$, then $X\times Y$ is null-additive in ${\mathbb{R}}^2$? Explicitly, must \[ \forall N\subseteq{\mathbb{R}}^2,\qquad N\text{ null}\ \Longrightarrow\ (X\times Y)+N\text{ null} \] hold under these hypotheses?

The supplied background asserts that both corresponding product implications hold for subsets of $\{0,1\}^{\mathbb N}$, with the product regarded as a subset of $\{0,1\}^{\mathbb N}\times\{0,1\}^{\mathbb N}$. The conventional interpretation uses coordinatewise addition modulo $2$, the product topology, and the fair-coin product probability measure, together with their product structures. The supplied excerpt does not explicitly specify these structures or define additivity; the definitions above and this interpretation therefore require confirmation against the original source. The registry labels the real-line questions open; this is a supplied status, not an independently verified current assessment.

Short English statement

Must the Cartesian product of any two meager-additive subsets of ${\mathbb{R}}$ be meager-additive in ${\mathbb{R}}^2$? Ask the same question separately for two null-additive subsets. The supplied source reports affirmative analogues for Cantor space and presents the real-line cases as open.

Sources

• Local UnsolvedMath registry: permanent ID $800626$, source code catalogue entry, title "Selection Principles in Mathematics --- Problem 55 ([prods])." The supplied record belongs to AMR Open Problem Lists and has upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82. Its supplied status is open.
• Tsaban, Selection Principles and Special Sets of Reals: Open Problems (2006), Problem 55. Supplied source URL: https://arxiv.org/abs/math/0609601. The registry attributes the problem to Tsaban and gives proposed year 2006; this does not establish original authorship of the question.
• Supplied provenance reports access on 2026-10-16 and extraction by arxiv_latex. This access date and the source assertions are supplied provenance, not independent verification performed for this draft.
• Supplied attribution/context: [prods]. The corresponding bibliographic entry is absent from the supplied record.