The Duffin-Schaeffer conjecture states that if $\psi: \Bbb R_+ \rightarrow \Bbb R_+$ is some function with $\sum (\varphi(n) \psi(n)/n)^k = \infty$, then the set of points $(x_1, \dots, x_k) \in [0,1]^k$ for which the system of inequalities $$ \left\vert x_i - {{p_i} \over {q}}\right\vert < {{\psi(q)}\over q}, \tag $*$ $$ has infinitely many integer solutions $(p_1, \dots, p_k) \in \Bbb Z^k$ and $q \in \Bbb N$ with $(p_i,q) = 1$ for $1 \leq i \leq k$ is full with respect to the Lebesgue measure on $\Bbb R^n$. Here $\varphi(n)$ denotes the Euler totient function of $n$. The conjecture has been established for $k \geq 2$ by {\it A. D. Pollington} and {\it R. C. Vaughan} [Mathematika 37, No. 2, 190--200 (1990;

Zbl 0715.11036)] and in the special case when $\psi$ is assumed to be non-increasing by {\it 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 $f$ with $x^{-k}f(x)$ monotonic, if $\sum f(\psi(n)/n) \varphi(n)^k = \infty$ then the Hausdorff $f$-measure of the set defined by (*) above is equal to the Hausdorff $f$-measure of $[0,1]^k$. As an immediate corollary, it is derived that the Hausdorff $f$-measure analogue of the Duffin-Schaeffer conjecture holds true for $k \geq 2$. 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 $f$-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.