Treffer: On Hilbert Cubes in Certain Sets: On Hilbert cubes in certain sets

Given \(k\geq 1\), a set \(H\subset \mathbb{N}\) is called a cube of size \(k\) if there exist \(a>0\) and \(x_1,\dots,x_k\) such that \(H=\{a+\sum_{i=1}^k\varepsilon_ix_i:\varepsilon=0\text{ or }1\}\). If the set of \(x\)'s is infinite the cube is called infinite. If \(\mathcal{A}\subset \mathbb{N}\) and \(n\in\mathbb{N}\) then \(F_{\mathcal{A}}(n)\) denotes the size of the largest Hilbert cube contained in \(\mathcal{A}\cap\{1,\dots,n\}\). Finally, let \(\mathcal{S}\), \(\mathcal{P}\), and \(\mathcal{P}_k\) denote the set of all squares, the set of all primes, and the set of positive integers composed of the primes not exceeding \(k\), respectively. The authors prove that (1) for \(n>n_0\) we have \(F_{\mathcal{S}}(n)0\), \(n>n_0(\varepsilon)\) we have \((1.1-\varepsilon)\log\log n0\), \(p>p_0(\varepsilon)\) we have \((1.1-\varepsilon)\log\log p