On Hilbert Cubes in Certain Sets

A set of the form {d + Σi εi a : εi = 0 or 1, Σi εi < ∞} (where d is a non-negative integer, a1, a2, ... are positive integers) is called a Hilbert cube. If {a1, a2, ...} is a finite set of, say, k elements, then it called a k-cube, while if {a1, a2, ...} is infinite, then the cube is said to be an infinite cube. As a partial answer to a question of Brown, Erdös and Freedman, an upper bound is given for the size of a Hilbert cube contained in the set of the squares not exceeding n. Estimates of Gaussian sums, Gallagher's “large sieves” and a result of Olson play a crucial rule in the proof. Hilbert cubes in other special sets are also studied.