Additive properties of dense subsets of sifted sequences
The authors begin by defining a sufficiently sifted sequence \({\mathcal A}\) of integers with dimension \(\kappa\). \({\mathcal A}^*\subset{\mathcal A}\) is called a dense subsequence of \({\mathcal A}\) if \(A^*(X)\geq A(X)/k\) for \(X\geq X_1\) and fixed \(k\geq 1\), where \(A(X)\) denotes the counting function of \({\mathcal A}\). This paper is concerned with establishing in Corollary 1 that, under certain conditions, \({\mathcal A}^*\) is an additive asymptotic basis of the integers and obtaining an upper bound of its order of the shape \(O_{\mathcal A}(k(\log\log 3k)^\kappa)\). For \({\mathcal A}\) the sequence of primes, when \(\kappa=1\), the order of the corresponding basis \({\mathcal P}^*\) obtained by \textit{A. Sárközy} [Astérisque 258, 109-127 (1999; Zbl 0969.11003)] was \(O(k^4)\). The above results follow from the authors' main theorem in which an asymmetrical lower bound for \(({\mathcal A}_1^*+{\mathcal A}_2^*)(X)\) is found, where, for \(i=1,2\), \({\mathcal A}_i\) is sufficiently sifted with dimension \(\kappa_i\), \({\mathcal A}_i^*\subset {\mathcal A}_i\) and \(A_i^*(X)\geq A_i(X)/k_i\) for \(X\geq X_1\) and fixed \(k_i\geq 1\). The authors go on to consider a finite version of these problems for which additional notation is required; we omit the details. The proofs, which are very technical although classed as elementary, depend on a version of the Selberg sieve and of the large sieve discussed in section 4, and then the required consequences are derived.
scientific article; zbMATH DE number 1770786
scientific article