Duffin–Schaeffer conjecture - Alchetron, the free social encyclopedia

The Duffin–Schaeffer conjecture is an important conjecture in metric number theory proposed by R. J. Duffin and A. C. Schaeffer in 1941. It states that if f : N → R + is a real-valued function taking on positive values, then for almost all α (with respect to Lebesgue measure), the inequality

| α − p q | < f ( q ) q

has infinitely many solutions in co-prime integers p , q with q > 0 if and only if the sum

∑ q = 1 ∞ f ( q ) φ ( q ) q = ∞ .

Here φ ( q ) is the Euler totient function.

The full conjecture remains unsolved. However, a higher-dimensional analogue of this conjecture has been resolved.

Progress

The implication from the existence of the rational approximations to the divergence of the series follows from the Borel–Cantelli lemma. The converse implication is the crux of the conjecture. There have been many partial results of the Duffin–Schaeffer conjecture established to date. Paul Erdős established in 1970 that the conjecture holds if there exists a constant c > 0 such that for every integer n we have either f ( n ) = c / n or f ( n ) = 0 . This was strengthened by Jeffrey Vaaler in 1978 to the case f ( n ) = O ( n − 1 ) . More recently, this was strengthened to the conjecture being true whenever there exists some ϵ > 0 such that the series ∑ n = 1 ∞ ( f ( n ) n ) 1 + ϵ φ ( n ) = ∞ . This was done by Haynes, Pollington, and Velani.

In 2006, Beresnevich and Velani proved that a Hausdorff measure analogue of the Duffin–Schaeffer conjecture is equivalent to the original Duffin–Schaeffer conjecture, which is a priori weaker. This result is published in the Annals of Mathematics.

References

Duffin–Schaeffer conjecture Wikipedia

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