A characterization of the linear sets satisfying Herz’s criterion

Haskell Paul Rosenthal

Vol. 28 (1969), No. 3, 663–668

DOI: 10.2140/pjm.1969.28.663

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)), if there exists an infinite set of positive integers N, such that

(*) for all integers j with 0 ≦ j < N, each of the numbers j∕N either belongs to E or is distant by at least 1∕N from E.

The main theorem proved here, is that E satisfies (H) if and only if there exists a sequence of sets F₁,F₂,⋯ with E = ⋂ _i=1^∞F_i and positive integers N₁ < N₂ < ⋯ satisfying the following properties for all i:

(1) N_i divides N_i+1 and F_i ⊃ F_i+1.

(2) F_i is a finite union of disjoint closed intervals each of whose end points is of the form j∕N_i for some integer j.

(3) If for some integer j, j∕N_i ∈ F_i, then j∕N_i ∈ F_i+1.

Mathematical Subject Classification

Primary: 26.10

Secondary: 42.00

Milestones

Received: 8 January 1968

Published: 1 March 1969