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Original Articles

Empirical Approach to the ×2, ×3 Conjecture

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Pages 252-268 | Published online: 20 May 2019
 

Abstract

We study atomic measures on [0,1] which are invariant both under multiplication by 2mod1 and by 3mod1, since such measures play an important role in deciding Furstenberg’s ×2,×3 conjecture. Our specific focus was finding atomic measures whose supports are far from being uniformly distributed, and we used computer software to discover a number of such measures (which we call outlier measures). The structure of these measures indicates the possibility that a sequence of atomic measures may converge to a non-Lebesgue measure; likely one which is a combination of the Lebesgue measure and one or more atomic measures.

Acknowledgments

Research of both authors is supported from:

  • Resources for science in years 2013–2018 as research project (NCN grant 2013/08/A/ST1/00275, Poland

  • Statutory research funds of Faculty of Pure and Applied Mathematics at Wrocław University of Science and Technology

Notes

1 The original conjecture deals more generally with T(x)=pxmod1 and S(x)=qxmod1 with p, q being a pair of coprime natural numbers; the pair (2, 3) is the simplest such pair.

Additional information

Funding

This study was supported by Narodowe Centrum Nauki (2013/08/A/ST1/00275) and Politechnika Wrocławska (0401/0155/18).

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