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 such as the squares, powerful numbers and pure powers.

19 pages; to appear in Revista Matemática Iberoamericana; this paper supersedes the earlier preprint arXiv:1207.1461