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.