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Hilbert cubes in arithmetic sets

    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
  • Texto completo no disponible (Saber más ...)
  • 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.


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