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Zbl 1148.11033
Beresnevich, Victor; Velani, Sanju
A mass transference principle and the Duffin-Schaeffer conjecture for Hausdorff measures. (English)
[J] Ann. Math. (2) 164, No. 3, 971-992 (2006). ISSN 0003-486X; ISSN 1939-0980/e

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)]. \par 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, Jarn\'\i{}k's Theorem is shown to be a consequence of Khintchine's Theorem together with the main result of the present paper. \par 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.
[Simon Kristensen (Aarhus)]

MSC 2000:: *11J83 Metric theory of numbers
11J13 Simultaneous homogeneous approximation, linear forms
28A78 Hausdorff measures
11H60 Mean value and transfer theorems

Keywords: Diophantine approximation; simultaneous approximation; Hausdorff measures

Citations: Zbl 0715.11036; JFM 52.0183.02

Cited in: Zbl 1236.11064 Zbl 1153.11035 Zbl 1129.11031

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