A mass transference principle and the Duffin-Schaeffer conjecture for Hausdorff measures. (English) Zbl 1148.11033

The Duffin-Schaeffer conjecture states that if \(\psi: \mathbb R_+ \rightarrow \mathbb R_+\) 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 \[ \left| x_i -\frac{p_i}{q}\right| < \frac{\psi(q)}{q}, \tag{*} \]
has infinitely many integer solutions \((p_1, \dots, p_k) \in \mathbb Z^k\) and \(q \in \mathbb N\) with \((p_i,q) = 1\) for \(1 \leq i \leq k\) is full with respect to the Lebesgue measure on \(\mathbb R^n\). Here \(\varphi(n)\) denotes the Euler totient function of \(n\). The conjecture has been established for \(k \geq 2\) 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 \(x^{-k}f(x)\) 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.


11J83 Metric theory
11J13 Simultaneous homogeneous approximation, linear forms
28A78 Hausdorff and packing measures
11H60 Mean value and transfer theorems
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