The density of \(B_ h[g]\) sequences and the minimum of dense cosine sums
A set of integers is called a \(B_h[ g]\) set if every integer has at most \(g\) representations as a sum of \(h\) elements of this set. The paper gives an upper bound for the maximal \(B_h[ 1]\) sequence contained in \([1, n]\). This bound is obtained via the method of exponential sums from the following inequality. If \(f(x)= M+ \sum^N_1 \cos \lambda_j x\) is a nonnegative cosine polynomial with frequencies \(1\leq \lambda_1< \dots< \lambda_N\leq (2- \varepsilon) N\), then \(M> C\varepsilon^2 N\) with some positive constant \(C\). The same bound was found with a combinatorial proof by \textit{X. Jia} [J. Number Theory 44, 84-92 (1993; Zbl 0778.11006)]. The second half of the paper is devoted to the construction of dense \(B_2[ 2]\) sequences. Such a sequence \(\subset [1, n]\) is found with \(\sim \sqrt {2n}\) elements, and an infinite sequence \(\{n_j\}\) is constructed with \(\liminf n_j/ j^2=1\).
scientific article; zbMATH DE number 844637