Piecewise syndetic sets in \(\mathbb{N}^t\) and \(\mathbb{N}^X\) (Q6542777)

Let \(\mathbb{N}\) be the set of positive integers, \([0, d]\) be the interval \(\{0, 1, 2, \ldots , d\}\) and also let \(S\) be an infinite subset of \(\mathbb{N}\). If there exists \(d\in \mathbb{N}\) such that \(S+[0, d]\) contains an infinite interval, then \(S\) is called syndetic. If there exists \(d\in \mathbb{N}\) such that \(S+[0, d]\) contains arbitrarily large finite intervals of \(\mathbb{N}\), then \(S\) is called piecewise syndetic. In this paper, the author gives simple proofs establishing that the basic facts concerning piecewise syndetic subsets of \(\mathbb{N}\) apply equally well to piecewise syndetic subsets of \(\mathbb{N}^t\), \(t\geq 1\). Some of these facts apply also to piecewise syndetic subsets of \(\mathbb{N}^X\) for any infinite set \(X\).