Diophantine approximation by algebraic hypersurfaces and varieties

Schmidt, Wolfgang

doi:10.1090/S0002-9947-06-04014-1

Diophantine approximation by algebraic hypersurfaces and varieties
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Trans. Amer. Math. Soc. 359 (2007), 2221-2241 Request permission

Abstract:

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.

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