By sum of two sets I mean $A+B := \{x+y:x \in A \quad y \in B\}$, and there is a tip in a book of real analysis by Zhou Minqiang which says:


“If $A,B$ are Borel sets in $\mathbb{R}^{n}$, $A+B$ may not be a Borel set.”


I want to know some specific examples.(Maybe $\mathbb{R}^{1}$ ?)


Any comments will be helpful.



An example that the sum of two Borel sets which is not a Borel set in n-dimensional Euclidean space
This is a result of Erdos and Stone: <https://www.ams.org/journals/proc/1970-025-02/S0002-9939-1970-0260958-1/S0002-9939-1970-0260958-1.pdf>


