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- W1872497211 abstract "We give a necessary and sufficient condition for the following property of an integer $dinmathbb N$ and a pair $(a,A)inmathbb R^2$: There exist $kappa > 0$ and $Q_0inmathbb N$ such that for all $mathbf xin mathbb R^d$ and $Qgeq Q_0$, there exists $mathbf p/qinmathbb Q^d$ such that $1leq qleq Q$ and $|mathbf x - mathbf p/q| leq kappa q^{-a} Q^{-A}$. This generalizes Dirichlet's theorem, which states that this property holds (with $kappa = Q_0 = 1$) when $a = 1$ and $A = 1/d$. We also analyze the set of exceptions in those cases where the statement does not hold, showing that they form a comeager set. This is also true if $mathbb R^d$ is replaced by an appropriate space, such as a nonsingular rational quadratic hypersurface which contains rational points. Finally, in the case $d = 1$ we describe the set of exceptions in terms of classical Diophantine conditions." @default.
- W1872497211 created "2016-06-24" @default.
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- W1872497211 date "2015-03-07" @default.
- W1872497211 modified "2023-09-27" @default.
- W1872497211 title "Variations on Dirichlet's theorem" @default.
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- W1872497211 hasPublicationYear "2015" @default.
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