Diophantine approximation for products of linear maps — logarithmic improvements

Gorodnik, Alexander; Vishe, Pankaj

doi:10.1090/tran/6953

Diophantine approximation for products of linear maps — logarithmic improvements
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by Alexander Gorodnik and Pankaj Vishe PDF

Trans. Amer. Math. Soc. 370 (2018), 487-507 Request permission

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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