There exist ≫ N^{1/8} Hilbert cubes of dimension 3 inside the squares with parameters ≤ N, and the ratios a_i/a_j for such cubes are dense in the positive reals.
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There are infinitely many Hilbert cubes of dimension 3 in the set of squares
There exist ≫ N^{1/8} Hilbert cubes of dimension 3 inside the squares with parameters ≤ N, and the ratios a_i/a_j for such cubes are dense in the positive reals.