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Diophantine property in the group of affine transformations of the li

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Abstract

We investigate the Diophantine property of a pair of elements in the group of affine transformations of the line. We say that a pair of elements γ1, γ2 in this group is Diophantine if there is a number A such that a product of length l of elements of the set {γ1, γ2, γ1-1,γ2-1} is either the unit element or of distance at least A-l from the unit element. We prove that the set of non-Diophantine pairs in a certain one parameter family is of Hausdorff dimension 0.

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Acknowledgement

I am greatly indebted to Menny Aka and Emmanuel Breuillard for communicating the problem to me. I thank them and Elon Lindenstrauss, Lior Rosenzweig, and Nicolas de Saxcé for stimulating discussions about this project.

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Authors and Affiliations

  1. Centre for Mathematical Sciences, Wilberforce Road, Cambridge, CB3 0WA, England

    Péter P. Varjú

  2. The Einstein Institute of Mathematics, Edmond J. Safra Campus, Givat Ram, The Hebrew University of Jerusalem, Jerusalem, 91904, Israel

    Péter P. Varjú

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  1. Péter P. Varjú
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Correspondence to Péter P. Varjú.

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Communicated by V. Totik

The support of the Simons Foundation and the European Research Council (Advanced Research Grant 267259) are gratefully acknowledged.

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Varjú, P.P. Diophantine property in the group of affine transformations of the li. ActaSci.Math. 80, 447–458 (2014). https://doi.org/10.14232/actasm-013-757-6

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  • DOI: https://doi.org/10.14232/actasm-013-757-6

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