On more rapid convergence to a density

Abstract. Let the set \(A\subset{\mathbb N}\) have positive asymptotic density \(d\) and the set \(|A(n)-nd|\) be not bounded above. Then for any \(d'\in(0,d)\) there exists a \(B\subset A\), such that the asymptotic density of \(B\) is \(d'\) and for infinitely many \(n\) we have \(|B(n)n^{-1}-d'|\) tends to zero more rapidly than \(|A(n)n^{-1}-d|\). This solves an open question of Rita Giuliano at al. [1].