This is not an answer to Muranov's problem. Below I present some partial results which can be interesting or helpful for researchers that will try to attack this problem in future. The proofs of the following statements can be found in this paper.

**Theorem 1.** Let $A,B$ be compact subsets of a topological group $G$ such that $aA\cup bB\subset A\cup B$ and $aA\cap bB\subset c(A\cap B)$ for some elements $a,b,c\in G$. The equalities $aA\cup bB=A\cup B$ and $aA\cap bB=c(A\cap B)$ hold if either the subgroup $H_3$ generated by the set $\{a,b,c\}$ is discrete or for some set $T\subset \{a,b,c\}$ with $\{a,b\}\subset T$, $\{a,c\}\subset T$ or $\{b,c\}\subset T$ the subgroup $H_2$ generated by $T$ is discrete and closed in $G$ and $H_2$ is normal in $H_3$.

This Theorem implies

**Corollary 1.** Let $A,B$ be compact subsets of a topological group $G$ such that $aA\cup bB\subset A\cup B$ and $aA\cap bB=\emptyset$ for some elements $a,b\in G$. The equalities $aA\cup bB=A\cup B$ and $A\cap B=\emptyset$ hold if either the subgroup $H_2$ generated by the set $\{a,b\}$ is discrete or for some non-empty set $T\subset\{a,b\}$ the subgroup $H_1$ generated by $T$ is discrete and closed in $G$ and $H_1$ is normal in $H_2$.

These results (and the original problem of Muranov) motivate the following definitions.

**Definition.** A topological group $G$ is called

*Muranov* if for any compact subsets $A,B\subset G$ and points $a,b,c\in G$ the inclusions $aA\cup bB\subset A\cup B$ and $aA\cap bB\subset c(A\cap B)$ imply the equalities $aA\cup bB=A\cup B$ and $aA\cap bB=c(A\cap B)$;
*weakly Muranov* if for any compact subsets $A,B\subset G$ and points $a,b\in G$ with $aA\cup bB\subset A\cup B$ and $aA\cap bB=\emptyset$ we get the equalities $aA\cup bB=A\cup B$ and $A\cap B=\emptyset$.

Theorem 1 and Corollary 1 imply

**Corollary 2.** An (abelian) topological group $G$ is

- Muranov if each 3-generated (resp. 2-generated) subgroup of $G$ is discrete;
- weakly Muranov if each 2-generated (resp. 1-generated) subgroup of $G$ is discrete.

**Corollary 3.** For every $n\in\mathbb N$ the topological group $\mathbb Q^n$ is Muranov and $\mathbb R^n$ is weakly Muranov.

**Corollary 4.** Each locally finite topological group is Muranov.

However the original problem of Muranov remains wide open:

**Problem.** Is the real line $\mathbb R$ Muranov? Is the circle $\mathbb R/\mathbb Z$ weakly Muranov?

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