DIRICHLET IMPROVABILITY FOR <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core=
Abstract
<jats:p> We study the problem of improving Dirichlet’s theorem of metric Diophantine approximation in the most general and multiplicative form in the <jats:inline-formula> <jats:alternatives> <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mi>S</mml:mi> </mml:math> <jats:tex-math>$S$</jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" content-type="simple" xlink:href="S1446788726101669_inline2.png"> <jats:alt-text content-type="machine-generated">upper S</jats:alt-text> </jats:inline-graphic> </jats:alternatives> </jats:inline-formula> -adic setting. Our approach is based on translation of the problem related to Dirichlet improvability into a dynamical one, and the main technique of our proof is the <jats:inline-formula> <jats:alternatives> <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mi>S</mml:mi> </mml:math> <jats:tex-math>$S$</jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" content-type="simple" xlink:href="S1446788726101669_inline3.png"> <jats:alt-text content-type="machine-generated">upper S</jats:alt-text> </jats:inline-graphic> </jats:alternatives> </jats:inline-formula> -adic version of quantitative nondivergence estimate due to Kleinbock and Tomanov. The main result of this paper can be regarded as the number field version of earlier works of Kleinbock and Weiss [‘Dirichlet’s theorem on Diophantine approximation and homogeneous flows’, <jats:italic>J. Mod. Dyn.</jats:italic> <jats:bold>4</jats:bold> (2008), 43–62], and of the second named author and Ghosh [‘Dirichlet’s theorem in function fields’, <jats:italic>Canad. J. Math.</jats:italic> <jats:bold>69</jats:bold> (3) (2017), 532–547]. Also, this in turn generalizes a result of Datta and Radhika [‘Singular Vectors on Manifolds over Number Fields’, <jats:italic>Monatsh. Math.</jats:italic> <jats:bold>200</jats:bold> (3) (2023), 545–568] on singularity of vectors to any number field <jats:inline-formula> <jats:alternatives> <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mi>K</mml:mi> </mml:math> <jats:tex-math>$K$</jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" content-type="simple" xlink:href="S1446788726101669_inline4.png"> <jats:alt-text content-type="machine-generated">upper K</jats:alt-text> </jats:inline-graphic> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mi>S</mml:mi> </mml:math> <jats:tex-math>$S$</jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" content-type="simple" xlink:href="S1446788726101669_inline5.png"> <jats:alt-text content-type="machine-generated">upper S</jats:alt-text> </jats:inline-graphic> </jats:alternatives> </jats:inline-formula> containing all Archimedian places. </jats:p>