Open Problems in Topological Algebra — Problem 11 (Banakh)

Definitions and mathematical statement

For subsets $B,C\subseteq{\mathbb{R}}$, define their sum and difference by \[ B+C=\{b+c:b\in B,\ c\in C\},\qquad B-C=\{b-c:b\in B,\ c\in C\}. \] All neighborhoods and interiors in the following question are taken in the usual topology of ${\mathbb{R}}$. For $A\subseteq{\mathbb{R}}$, write \[ 4A=A+A+A+A=\{a_1+a_2+a_3+a_4:a_1,a_2,a_3,a_4\in A\}, \] where the four elements need not be distinct.

Question (Banakh; supplied Problem 11). Is it true that for every compact set $A\subseteq{\mathbb{R}}$, \[ \left(\exists\varepsilon>0\;(-\varepsilon,\varepsilon)\subseteq A-A\right) \quad\Longrightarrow\quad \operatorname{int}_{{\mathbb{R}}}(4A)\ne\varnothing? \] Equivalently, under the displayed hypothesis, must there exist $x\in{\mathbb{R}}$ and $\delta>0$ such that \[ (x-\delta,x+\delta)\subseteq A+A+A+A? \]

Unresolved contextual material. The supplied statement begins with a paragraph about an unspecified group: it says that such a group, if it exists, cannot be both analytic and abelian, and gives the more general exclusion involving a SIN-group. The paragraph defines an analytic space as a metrizable continuous image of a separable complete metric space. It also asserts, citing only [BZ], that $\mathbb{Z}^{\omega}$ contains a dense $o$-bounded $G_\delta$-subset on which the first player has a winning strategy in the game $\mathrm{OF}$. The group referred to by "such a group," the game, the meaning of $o$-boundedness, and the bibliographic identity of [BZ] are not supplied. No connection between that paragraph and the compact-set question is given. These are recorded as unresolved source context and are not imposed as additional hypotheses on $A$.

Short English statement

Must the sum of four copies of a compact subset $A$ of the real line contain a nonempty open interval whenever $A-A$ contains an open interval around zero? The supplied registry labels this question open; that status has not been independently verified. An accompanying paragraph about topological groups has an unresolved referent and relationship to this question.

Sources

• Local UnsolvedMath registry, supplied complete source record with permanent ID $800737$, exact problem code catalogue entry, and title "Open Problems in Topological Algebra --- Problem 11 (Banakh)." The record lists Banakh and Choban as proposers, gives the year 2012, and labels the problem open. These are supplied registry attributions and status labels, not independently verified findings.
• Banakh and Choban, Some Open Problems in Topological Algebra (2012), Problem 11, as identified by the supplied provenance: https://arxiv.org/abs/1202.1619. The source notes specifically attribute the question/context to Banakh. The supplied access date is 2026-10-16 and the extraction method is arxiv_latex; these provenance details are reproduced without claiming independent source access or verification.
• The preliminary contextual paragraph cites [BZ] for its assertion about a dense $o$-bounded $G_\delta$-subset of $\mathbb{Z}^{\omega}$ and the game $\mathrm{OF}$. No bibliographic entry or link identifying [BZ] is included in the supplied record.