The Duffin-Schaeffer conjecture states that if is some function with
, then the set of points for which the system of inequalities
has infinitely many integer solutions and with for is full with respect to the Lebesgue measure on . Here denotes the Euler totient function of . The conjecture has been established for by A. D. Pollington and R. C. Vaughan [Mathematika 37, No. 2, 190–200 (1990; Zbl 0715.11036)] and in the special case when is assumed to be non-increasing by A. Khintchine [Math. Z. 24, 706–714 (1926; JFM 52.0183.02)].
In the present important paper, the authors establish that if the Duffin-Schaeffer conjecture is true, then a similar seemingly stronger statement for general Hausdorff measures is also true. More precisely, if the Duffin-Schaeffer conjecture holds, then for any dimension function with monotonic, if then the Hausdorff -measure of the set defined by (*) above is equal to the Hausdorff -measure of . As an immediate corollary, it is derived that the Hausdorff -measure analogue of the Duffin-Schaeffer conjecture holds true for . Also, Jarník’s Theorem is shown to be a consequence of Khintchine’s Theorem together with the main result of the present paper.
The main tool underlying the proof of the above results is the so-called Mass Transference Principle, which is applicable to a much wider setup than that of Duffin-Schaeffer type problems. This result gives a way of transfering results about the Lebesgue measure of a limsup set to results about general Hausdorff -measures of related limsup sets. Applying this principle to the particular limsup sets defined by (*) yields the above results. In addition to Euclidean space, the method is also valid for a large class of locally compact metric spaces. The proof of the Mass Transference Principle relies on an intricate Cantor set construction.