Motivation. Recently a group of researchers found an aperiodic monotile in $\mathbb{R}^2$, answering a long-standing question. There are many results in higher dimensions, so let's explore the lower dimension.

Formalisation. For any set $S\subseteq \mathbb{R}$ and $a\in \mathbb{R}$, we set $a+S = \{a+s: s\in S\}$, and for any collection of subsets ${\frak S}\subseteq {\cal P}(\mathbb{R})$ we let $a + {\frak S} = \{a+S: S\in{\frak S}\}$.

We say that a partition ${\frak T}$ of $\mathbb{R}$ is a mono-tiling if for any $T_0\neq T_1\in {\frak T}$ there is $a\in\mathbb{R}$ such that $T_0 = a + T_1$. We call the mono-tiling ${\frak T}$ aperiodic if for all $a\in \mathbb{R}$ we have ${\frak T}\neq a + {\frak T}$.

Question. Does $\mathbb{R}$ have an aperiodic mono-tiling?

Motivation. Recently a group of researchers found an aperiodic monotile in $\mathbb{R}^2$, answering a long-standing question. There are many results in higher dimensions, so let's explore the lower dimension.

Formalisation. For any set $S\subseteq \mathbb{R}$ and $a\in \mathbb{R}$, we set $a+S = \{a+s: s\in S\}$, and for any collection of subsets ${\frak S}\subseteq {\cal P}(\mathbb{R})$ we let $a + {\frak S} = \{a+S: S\in{\frak S}\}$.

We say that a partition ${\frak T}$ of $\mathbb{R}$ is a mono-tiling if for any $T_0\neq T_1\in {\frak T}$ there is $a\in\mathbb{R}$ such that $T_0 = a + T_1$. We call the mono-tiling ${\frak T}$ aperiodic if for all $y\in \mathbb{R}$ we have ${\frak T}\neq y + {\frak T}$.

Question. Does $\mathbb{R}$ have an aperiodic mono-tiling?