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On the Asymptotic Density of Sets of Integers
Author(s) -
Saffari B.
Publication year - 1976
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/s2-13.3.475
Subject(s) - citation , mathematics , combinatorics , computer science , arithmetic , library science , discrete mathematics
where a,, and b, equal 0 or 1, the set A (resp. B) appearing as the set of those n such that a,, = 1 (resp. b, = 1). It seems a rather difficult problem to describe explicitly the structure of all such direct factors, although the theorem demonstrated in [5] and our present Theorem 1 shed some light on the situation by proving the existence of their asymptotic densities. (The corresponding additive problem for N is much easier, and has been completely settled by de Bruijn [l] in 1956 and by Long [4] in 1967.) Although this structure problem for A and B, as pointed out in [5; $9.31, essentially reduces to an algebraic and combinatorial problem, it has close connections with sets of multiples (see [3; Chapter V]), and also has “ multiplicative ” features, in the (somewhat extended) sense of multiplicative functions. These two interesting aspects can already be foreseen from (respectively) the proof of our present Theorem 2 and the new proof of our result given in [2], but will be discussed further elsewhere.