Hilbert cubes in arithmetic sets

  • Rainer Dietmann

    Royal Holloway University of London, Egham, UK
  • Christian Elsholtz

    Technische Universität Graz, Austria


We show upper bounds on the maximal dimension dd of Hilbert cubes H=a0+{0,a1}++{0,ad}S[1,N]H=a_0+\{0,a_1\}+\cdots + \{0, a_d\}\subset S \cap [1, N] in several sets SS of arithmetic interest.

a) For the set of squares we obtain d=O(loglogN)d=O(\mathrm {log} \mathrm {log} N). 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 WW of powerful numbers we show d=O((logN)2)d=O((\mathrm {log} N)^2).

c) For the set VV of pure powers we also show d=O((logN)2)d=O((\mathrm {log} N)^2), but for a homogeneous Hilbert cube, with a0=0a_0=0, this can be improved to d=O((loglogN)3/logloglogN)d=O((\mathrm {log}\mathrm {log} N)^3/\mathrm {log} \mathrm {log} \mathrm {log} N), when the aia_i are distinct, and d=O((loglogN)4/(logloglogN)2)d=O((\mathrm {log} \mathrm {log} N)^4/(\mathrm {log} \mathrm {log} \mathrm {log} N)^2), generally. This compares with a result of d=O((logN)3/(loglogN)1/2)d = O((\mathrm {log} N)^3/(\mathrm {log} \mathrm {log} N)^{1/2}) in the literature.

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

e) For a set without arithmetic progressions of length kk we prove d=Ok(logN)d=O_k(\mathrm {log} N), which is close to the true order of magnitude.

Cite this article

Rainer Dietmann, Christian Elsholtz, Hilbert cubes in arithmetic sets. Rev. Mat. Iberoam. 31 (2015), no. 4, pp. 1477–1498

DOI 10.4171/RMI/877