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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Diophantine approximation for products of linear maps -- logarithmic improvements


Authors: Alexander Gorodnik and Pankaj Vishe
Journal: Trans. Amer. Math. Soc. 370 (2018), 487-507
MSC (2010): Primary 11D75, 11J20, 11K60, 37A17, 37A45
DOI: https://doi.org/10.1090/tran/6953
Published electronically: June 21, 2017
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Abstract: This paper is devoted to the study of a problem of Cassels in multiplicative Diophantine approximation which involves minimising values of a product of affine linear forms computed at integral points. It was previously known that values of this product become arbitrary close to zero, and we establish that, in fact, they approximate zero with an explicit rate. Our approach is based on investigating quantitative density of orbits of higher-rank abelian groups.


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Additional Information

Alexander Gorodnik
Affiliation: School of Mathematics, University of Bristol, Bristol BS8 1SD, United Kingdom
Email: a.gorodnik@bristol.ac.uk

Pankaj Vishe
Affiliation: Department of Mathematics, Durham University, Durham DH1 3LE, United Kingdom
Email: pankaj.vishe@durman.ac.uk

DOI: https://doi.org/10.1090/tran/6953
Received by editor(s): January 14, 2016
Received by editor(s) in revised form: April 1, 2016, and April 7, 2016
Published electronically: June 21, 2017
Article copyright: © Copyright 2017 American Mathematical Society

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