Title: A characterization of the linear sets satisfying Herz’s criterion
Abstract:Let E be a closed subset of T, the circle group, which we identify with the real numbers modulo 1. E is said to satisfy Herz's criterion (briefly, E satisfies (H)) 9 if there exists an infinite set of...Let E be a closed subset of T, the circle group, which we identify with the real numbers modulo 1. E is said to satisfy Herz's criterion (briefly, E satisfies (H)) 9 if there exists an infinite set of positive integers N, such that (*) for all integers j with 0 ^ j < N, each of the numbers jIN either belongs to E or is distant by at least 1/JV from E.The main theorem proved here, is that E satisfies (H) if and only if there exists a sequence of sets F u F 2 , with E= Π^Fi and positive integers Λ7Ί < N 2 < ••• satisfying the following properties for all i:(1) Ni divides i SΓ»+i and F* => F i+ί .(2) Fi is a finite union of disjoint closed intervals each of whose end points is of the form j/Ni for some integer j.(3) If for some integer j,jlNieF i9 thenRead More