Hilbert cubes in arithmetic sets
Hilbert cubes in arithmetic sets
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 …