Vol. 76, No. 2, 1978

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On the metric theory of Diophantine approximation

Jeffrey D. Vaaler

Vol. 76 (1978), No. 2, 527–539
Abstract

A conjecture of Duffin and Schaeffer states that

∑∞         −1
αnφ(n)n  = + ∞
n=2

is a necessary and sufficient condition that for almost all real x there are infinitely many positive integers n which satisfy |x a∕n| < αnn1 with (a,n) = 1. The necessity of the condition is well known. We prove that the condition is also sufficient if αn = O(n1).

Mathematical Subject Classification
Primary: 10K10, 10K10
Milestones
Received: 29 April 1977
Published: 1 June 1978
Authors
Jeffrey D. Vaaler
Department of Mathematics
University of Texas at Austin
1 University Station - C1200
Austin TX 78712-0257
United States