On the Asymptotic Density of Sets of Integers

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.