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Hilbert cubes in progression-free sets and in the set of squares

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Abstract

We present a new method to give upper bounds on the dimension of Hilbert cubes in certain sets. As a special case we improve Hegyvári and Sárközy’s upper bound O((logN)1/3) for the maximal dimension of a Hilbert cube in the set of squares in [1,N] to O((log logN)2).

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Authors and Affiliations

  1. Department of Mathematics, Royal Holloway, University of London, Egham, TW20 0EX, Surrey, UK

    Rainer Dietmann

  2. Institut für Mathematik A, Technische Universität Graz, Steyrergasse 30, A-8010, Graz, Austria

    Christian Elsholtz

Authors
  1. Rainer Dietmann
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  2. Christian Elsholtz
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Correspondence to Rainer Dietmann.

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Dietmann, R., Elsholtz, C. Hilbert cubes in progression-free sets and in the set of squares. Isr. J. Math. 192, 59–66 (2012). https://doi.org/10.1007/s11856-012-0047-7

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  • DOI: https://doi.org/10.1007/s11856-012-0047-7

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