Some continuity properties of the Schnirelmann density

R. L. Duncan

Vol. 26 (1968), No. 1, 57–58

DOI: 10.2140/pjm.1968.26.57

Abstract

Let S denote the set of all infinite increasing sequences of positive integers. For all A = {a_n} and B = {b_n} in S, define the metric ρ(A,B) = 0 if A = B, i.e., if a_n = b_n for all n and ρ(A,B) = 1∕k otherwise, where k is the smallest value of n for which a_n≠b_n. Similar metrics have been considered previously [1, 21.

Our purpose here is to discuss several continuity properties of the Schnirelmann density d(A) = inf A(n)∕n, where A(n) is the number of elements of A not exceeding n. In particular, we obtain a characterization of the set of all sequences having density zero as the set of points of continuity of d(A).

Mathematical Subject Classification

Primary: 10.49

Milestones

Received: 18 July 1967

Published: 1 July 1968