The Duffin-Schaeffer conjecture is a metric criterion for approximation of real numbers by reduced rational fractions. For an arbitrary
function 
,
it asserts that divergence of
 |
implies that, for Lebesgue-almost every 
, there are infinitely many coprime integers 
and 
satisfying
 |
where 
is the totient function. Koukoulopoulos and Maynard
(2020) proved this assertion without any monotonicity assumption. The fixed-shift
problem with unrestricted numerators is the weak inhomogeneous Duffin-Schaeffer conjecture.
