Hilbert cubes in arithmetic sets

Abstract

We show upper bounds on the maximal dimension d of Hilbert cubes H=a_0+\{0,a_1\}+\cdots + \{0, a_d\}\subset S \cap [1, N] in several sets S of arithmetic interest. a) For the set of squares we obtain 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 W of powerful numbers we show d=O((\mathrm {log} N)^2) . c) For the set V of pure powers we also show d=O((\mathrm {log} N)^2) , but for a homogeneous Hilbert cube, with a_0=0 , this can be improved to d=O((\mathrm {log}\mathrm {log} N)^3/\mathrm {log} \mathrm {log} \mathrm {log} N) , when the a_i are distinct, and 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((\mathrm {log} N)^3/(\mathrm {log} \mathrm {log} N)^{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=O_k(\mathrm {log} N) , which is close to the true order of magnitude.

Locations

• Revista Matemática Iberoamericana - View - PDF
• arXiv (Cornell University) - View - PDF