An important consequence of Dirichlet’s theorem (1842) is that any
irrational number can be approximated by infinitely many rationals with an error of approximation one over denominator square. Surprisingly, most of the metrical results to date improve this corollary instead of Dirichlet’s original theorem.
Recently, metric theoretic results for the sets of real numbers admitting improvements to Dirichlet’s original theorem have been developed by Hussain, Kleinbock and others (2017, 2018).
Full abstract: PDF
This is a joint work with Mumtaz Hussain.
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