Duffin–Schaeffer conjecture

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 + {displaystyle f:mathbb {N} ightarrow mathbb {R} ^{+}} is a real-valued function taking on positive values, then for almost all α {displaystyle alpha } (with respect to Lebesgue measure), the inequality 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 + {displaystyle f:mathbb {N} ightarrow mathbb {R} ^{+}} is a real-valued function taking on positive values, then for almost all α {displaystyle alpha } (with respect to Lebesgue measure), the inequality has infinitely many solutions in co-prime integers p , q {displaystyle p,q} with q > 0 {displaystyle q>0} if and only if the sum Here φ ( q ) {displaystyle varphi (q)} is the Euler totient function. A higher-dimensional analogue of this conjecture was resolved by Vaughan and Pollington in 1990. In July 2019, D. Koukoulopoulos and James Maynard announced a proof of the conjecture. 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 {displaystyle c>0} such that for every integer n {displaystyle n} we have either f ( n ) = c / n {displaystyle f(n)=c/n} or f ( n ) = 0 {displaystyle f(n)=0} . This was strengthened by Jeffrey Vaaler in 1978 to the case f ( n ) = O ( n − 1 ) {displaystyle f(n)=O(n^{-1})} . More recently, this was strengthened to the conjecture being true whenever there exists some ϵ > 0 {displaystyle epsilon >0} such that the series ∑ n = 1 ∞ ( f ( n ) n ) 1 + ϵ φ ( n ) = ∞ {displaystyle sum _{n=1}^{infty }left({frac {f(n)}{n}} ight)^{1+epsilon }varphi (n)=infty } . 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. In July 2019, Dimitris Koukoulopoulos and James Maynard announced a proof of the conjecture.