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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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