A characterization of some additive arithmetical functions, III

I. Introduction. In 1946, P. Erdős [2] proved that if a real-valued additive arithmetical function f satisfies the condition: f(n+1) - f(n) → 0, n → ∞, then there exists a constant C such that f(n) = C log n for all n in ℕ*. Later, I. Kátai [3,4] was led to conjecture that it was possible to determine additive arithmetical functions f and g satisfying the condition: there exist a real number l, a, c in ℕ*, and integers b, d such that f(an+b) - g(cn+d) → l, n → ∞. This problem has been treated essentially by analytic methods ([1], [7]). In this article, we shall provide, in an elementary way, a characterization of real-valued additive arithmetical functions f and g satisfying the condition: (H) there exist a and b in ℕ* with (a,b) = 1 and a finite set Ω such that (1) lim_{n→∞} min_{ω∈Ω} |f(an+b) - g(n) - ω| = 0.