Let \(\mathbb Z\) denote the set of integers and let \(n\) be a positive integer. We say that a set \(A\subseteq \mathbb Z^n\) has upper density at least \(\varepsilon\) (where \(\varepsilon\) is a real number, \(0<\varepsilon\leq1),\) if there is a sequence of \(n\)-dimensional cubes \(B_{R_j}\) , \(j=1,2,\dots,\) of sizes \(R_j\rightarrow\infty,\) not necessarily centered at the origin, such that, for all \(j,\) \[ |A\cap B_{R_j}|\geq\varepsilon {R_j}^n. \] The main result of the paper is the following theorem.
Let \(n\geq5,\) \(1\geq\varepsilon>0\) and \(A\subseteq \mathbb Z^n\) having upper density at least \(\varepsilon.\) Then there exist a positive integer \(Q_\varepsilon,\) depending only on \(\varepsilon,\) and a positive integer \(\Lambda_A\) depending on the set \(A,\) such that, for every integer \(\lambda\geq\Lambda_A,\) \(\lambda Q_\varepsilon^2\) belongs to the set \(\{|m-\ell|^2~;~m\in A,\ell\in A\}.\) Here \(m=(m_1,\dots,m_n),\ell= (\ell_1,\dots,\ell_n),\) and \(|m-\ell|\) denotes the distance \(\sum_{i=1}^n|m_i-\ell_i|.\)
The author explains why such a result does not hold for \(n\leq 3\) and leaves open the case \(n=4.\)
The work is motivated by the papers [H. Furstenberg, Y. Katznelson and B. Weiss, Algorithms Comb. 5, 184–198 (1990; Zbl 0738.28013) and J. Bourgain, Isr. J. Math. 54, 307–316 (1986; Zbl 0609.10043)].