Hilbert cubes in arithmetic sets

• Rainer Dietmann ^[1] ; Christian Elsholtz ^[2]
  1. [1] University of London

     University of London

     Reino Unido

     
  2. [2] Graz University of Technology

     Graz University of Technology

     Graz, Austria

     
• Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 31, Nº 4, 2015, págs. 1477-1498
• Idioma: inglés
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• Resumen

  • We show upper bounds on the maximal dimension dd of Hilbert cubes H=a0+{0,a1}+⋯+{0,ad}⊂S∩[1,N] in several sets SS of arithmetic interest.

    a) For the set of squares we obtain d=O(loglogN). Using previously known methods this bound could have been achieved only conditionally subject to an unsolved problem of Erdős and Radó.

    b) For the set W of powerful numbers we show d=O((logN)2).

    c) For the set V of pure powers we also show d=O((logN)2), but for a homogeneous Hilbert cube, with a0=0, this can be improved to d=O((loglogN)3/logloglogN), when the aiai are distinct, and d=O((loglogN)4/(logloglogN)2), generally. This compares with a result of d=O((logN)3/(loglogN)1/2) in the literature.

    d) For the set V we also solve an open problem of Hegyvári and Sárközy, namely we show that V does not contain an infinite Hilbert cube.

    e) For a set without arithmetic progressions of length k we prove d=Ok(logN), which is close to the true order of magnitude.