On properties of some bent function classes constructed using subspaces

The closest bent functions to bent functions from the class $\mathcal{PS}_{ap}$ and its extensions are considered. Such bent functions are constructed using the affinity of a given bent function in $2n$ variables on $n$-dimensional affine subspaces of $\mathbb{F}_2^{2n}$ which is similar to the method of generating bent functions from the considered classes. It is proved that almost all bent functions from $\mathcal{PS}_{ap}$ (and its extensions) have exactly $2^{n - 1} + 1$ closest bent functions. In other words, they are affine on exactly $2^{n - 1} + 1$ distinct $n$-dimensional affine subspaces of $\mathbb{F}_2^{2n}$, all of which are used for constructing $\mathcal{PS}_{ap}$. We also prove that the number of distinct closest bent functions to the functions from $\mathcal{PS}_{ap}$ is equal to $3|\mathcal{PS}_{ap}| - o(|\mathcal{PS}_{ap}|)$ as $n \to \infty$. Almost all of them belong to the natural extension of $\mathcal{PS}_{ap}$.