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 Rn 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 Rn by algebraic hypersurfaces, or more generally algebraic varieties defined over the rationals.