On the Distribution of the Convergents of Almost All Real Numbers

Let n 1 < n 2 < … be an infinite sequence of integers. The necessary and sufficient condition that for almost all α the inequality |α − a i n i | < ϵ n i 2 with ( a i , n i ) = 1 should have infinitely many solutions is that Σ i=1 ∞ ϕ(n i) n i 2 = ∞ . The techniques used in the proof can perhaps be applied to prove an old conjecture of Duffin and Schaeffer.

Английский

Journal of Number Theory. – 1970. – Vol. 2, № 4. – P. 425–441.