A mass transference principle and the Duffin-Schaeffer conjecture for Hausdorff measures.

*(English)*Zbl 1148.11033The 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, JarnĂk’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.

\(\sum (\varphi(n) \psi(n)/n)^k = \infty\), 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, JarnĂk’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.

Reviewer: Simon Kristensen (Aarhus)