Selection Principles in Mathematics — Problem 56 ([prods])

Definitions and mathematical statement

Let $\mathbb N=\{1,2,\ldots\}$. For an abelian topological group $G$, let $\mathcal M(G)$ denote its meager subsets, where a set is meager if it is a countable union of nowhere dense sets. When a measure on $G$ is specified, let $\mathcal N(G)$ denote the sets of outer measure zero. For $\mathcal I\in\{\mathcal M,\mathcal N\}$, call $A\subseteq G$ $\mathcal I$-additive if \[ \forall B\subseteq G\quad \left(B\in\mathcal I(G)\Longrightarrow A+B\in\mathcal I(G)\right), \qquad A+B=\{a+b:a\in A,\ b\in B\}. \] For $G=\mathbb R$, use ordinary addition, the usual topology, and Lebesgue measure. The standard interpretation for $C=\{0,1\}^{\mathbb N}$ uses the product topology, coordinatewise addition modulo $2$, denoted by $\oplus$, and the product probability measure assigning probability $1/2$ to each coordinate value. These ambient-group and measure conventions make the usual meanings explicit; the supplied record does not itself define them.

The source asks whether meager-additivity and, separately, null-additivity transfer from $\mathbb R$ to $C$ for a set $X\subseteq[0,1]$. Its phrase "when viewed as a subset of $\{0,1\}^{\mathbb N}$" leaves the identification unspecified. To display the intended implications precisely conditional on a binary-representation choice, put \[ \pi:C\longrightarrow[0,1],\qquad \pi(s)=\sum_{n=1}^{\infty}s_n2^{-n}, \] and let $b:[0,1]\longrightarrow C$ be a fixed choice of binary expansion satisfying $\pi(b(x))=x$ for every $x\in[0,1]$. For this interpretation, the two questions are whether, for every $X\subseteq[0,1]$, respectively, \[ \begin{aligned} &\left[\forall M\in\mathcal M(\mathbb R),\ X+M\in\mathcal M(\mathbb R)\right] \Longrightarrow \left[\forall K\in\mathcal M(C),\ b(X)\oplus K\in\mathcal M(C)\right],\\ &\left[\forall N\in\mathcal N(\mathbb R),\ X+N\in\mathcal N(\mathbb R)\right] \Longrightarrow \left[\forall L\in\mathcal N(C),\ b(X)\oplus L\in\mathcal N(C)\right]. \end{aligned} \] Here $b(X)=\{b(x):x\in X\}$. The two implications are separate questions, not a joint hypothesis requiring both kinds of additivity. The record does not specify a binary-expansion convention, whether the intended realization is instead the full inverse image $\pi^{-1}(X)$, or a quantifier over representation choices. Thus the displayed formulation is conditional on the stated interpretation and requires source clarification before being identified with the exact intended formulation.

Short English statement

For a set $X\subseteq[0,1]$, does the property that adding any meager real set gives a meager set persist when $X$ is represented in the binary sequence group? Ask the same question separately for null sets. The representation and ambient-group conventions require confirmation. The registry labels this problem open; that status is supplied metadata, not an independently verified conclusion.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $800627$, exact source problem code catalogue entry, title "Selection Principles in Mathematics --- Problem 56 ([prods])." The record lists Tsaban as proposer, gives the year 2006, and labels the problem open.
• Supplied source attribution: Tsaban, Selection Principles and Special Sets of Reals: Open Problems (2006), Problem 56. Supplied URL: https://arxiv.org/abs/math/0609601. The background records access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance statements, not access or verification performed for this draft.
• The supplied attribution/context marker is [prods]. Its bibliographic expansion is absent from the record and has not been reconstructed. The supplied upstream revision is d371ee917ac9647fbb06cd6fc529deec6f3dda82.