In 2020, Parusinski and Rond proved that every algebraic set $V \subset \mathbb{R}^n$ is homeomorphic to a $\bar{\mathbb{Q}}^r$-algebraic set $V' \subset \mathbb{R}^n$, where $\bar{\mathbb{Q}}^r$ denotes the field of real algebraic numbers. Latter very general result motivates the following open problem: $\mathbb{Q}$-algebraicity problem: (Parusinski, 2021) Is every algebraic set $V \subset \mathbb{R}^n$ homeomorphic to some $\mathbb{Q}$-algebraic set $V' \subset \mathbb{R}^m$, with $m \ge n$? The aim of the talk is to introduce above open problem and to explain how our new approximation techniques over $\mathbb{Q}$ allowed us to provide some classes of real algebraic sets that positively answer the $\mathbb{Q}$-algebraicity problem.