\(B_ h[g]\)-sequences with large upper density
Let \(\mathbb{Z}\) be the set of all integers and \(2\leq h\in \mathbb{Z}\), \(1\leq g\in \mathbb{Z}\). A subset \(A\) of \(\mathbb{Z}\) is called a \(B_h[g]\)-sequence if for every positive integer \(m\) the equation \(m=x_1+x_2 + \cdots +x_h\), \(x_1<x_2< \cdots <x_h\), \(x_i\in A\) has at most \(g\) distinct solutions. The author constructs \(B_h[g]\)-sequences with large density. It is shown that for every \(g\geq 2\) there exists a finite \(B_2[g]\)-sequence \(B\), \(B\subseteq [1,n]\) with \[ |B|= \sqrt {2g-1} \sqrt n+o (\sqrt n) \] and an infinite \(B_2[g]\)-sequence \(\{n_1,n_2, \dots\}\), \(n_1<n_2 < \dots\) with \[ \liminf_{j\to\infty} {n_j\over j^2} = {1\over \sqrt {2g-1}}. \] In both cases finite \(B_2[1]\)-sequences constructed by the method of Bose and S. Chowla were modified (and extended) to \(B_2[g]\)-sequences with the stated properties. In the finite case this theorem improves for \(g=2\) a result of Kolountzakis who recently constructed a \(B_2[2]\)-sequence \(B \subseteq [1,n]\) with \(|B|\geq \sqrt {2n} + o(\sqrt n)\). Also given are results for finite \(B_h [g]\)-sequences and quasi-\(B_2[2]\)-sequences. Some related open problems are discussed.
\(B_ h[g]\)-sequences with large density
scientific article; zbMATH DE number 874181