Diophantine approximation by algebraic hypersurfaces and varieties

Questions on rational approximations to a real number can be generalized in two directions. On the one hand, we may ask about ``approximation'' to a point in $\mathbb{R}^{n}$ by hyperplanes defined over the rationals. That is, we seek hyperplanes with small distance from the given point. On the other hand, following Wirsing, we may ask about approximation to a real number by real algebraic numbers of degree at most $d$.

The present paper deals with a common generalization of both directions, namely with approximation to a point in $\mathbb{R}^{n}$ by algebraic hypersurfaces, or more generally algebraic varieties defined over the rationals.