Points near real algebraic sets

Abstract:

Given a real algebraic set $X$ and a box $\mathscr {B}$ in $\mathbb {R}^n$, which is a union of cubes of equal size and with disjoint interiors, we bound the number of cubes that intersect $X$. As a consequence, we bound the volume of the set of points having distance at most $\delta$ from $X \cap \mathscr {B}$, and we estimate the number of integer points in a domain $\mathscr {D} \subset \mathbb {R}^n$ bounded by algebraic hypersurfaces.