Vol. 28, No. 3, 1969

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A characterization of the linear sets satisfying Herz’s criterion

Haskell Paul Rosenthal

Vol. 28 (1969), No. 3, 663–668

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=1Fi and positive integers N1 < N2 < satisfying the following properties for all i:

(1) Ni divides Ni+1 and Fi Fi+1.

(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, j∕Ni Fi, then j∕Ni Fi+1.

Mathematical Subject Classification
Primary: 26.10
Secondary: 42.00
Received: 8 January 1968
Published: 1 March 1969
Haskell Paul Rosenthal