# On the distribution of the convergents of almost all real numbers

- Authors
- Journal
- Journal of Number Theory 0022-314X
- Publisher
- Elsevier
- Publication Date
- Volume
- 2
- Issue
- 4
- Identifiers
- DOI: 10.1016/0022-314x(70)90046-6

## Abstract

Abstract 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.

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