*(English)*Zbl 1148.11033

The Duffin-Schaeffer conjecture states that if $\psi :{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ is some function with

$\sum {(\varphi \left(n\right)\psi \left(n\right)/n)}^{k}=\infty $, then the set of points $({x}_{1},\cdots ,{x}_{k})\in {[0,1]}^{k}$ for which the system of inequalities

has infinitely many integer solutions $({p}_{1},\cdots ,{p}_{k})\in {\mathbb{Z}}^{k}$ and $q\in \mathbb{N}$ with $({p}_{i},q)=1$ for $1\le i\le k$ is full with respect to the Lebesgue measure on ${\mathbb{R}}^{n}$. Here $\varphi \left(n\right)$ denotes the Euler totient function of $n$. The conjecture has been established for $k\ge 2$ by *A. D. Pollington* and *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 *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 ${x}^{-k}f\left(x\right)$ monotonic, if $\sum f(\psi \left(n\right)/n)\varphi {\left(n\right)}^{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 $k\ge 2$. 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.