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

The Duffin-Schaeffer conjecture states that if ψ: + + is some function with

(ϕ(n)ψ(n)/n) k =, then the set of points (x 1 ,,x k )[0,1] k for which the system of inequalities

x i -p i q<ψ(q) q,(*)

has infinitely many integer solutions (p 1 ,,p k ) k and q with (p i ,q)=1 for 1ik is full with respect to the Lebesgue measure on n . Here ϕ(n) denotes the Euler totient function of n. The conjecture has been established for k2 by A. D. Pollington and R. C. Vaughan [Mathematika 37, No. 2, 190–200 (1990; Zbl 0715.11036)] and in the special case when ψ 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 f(ψ(n)/n)ϕ(n) k = 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 k2. 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.


MSC:
11J83Metric theory of numbers
11J13Simultaneous homogeneous approximation, linear forms
28A78Hausdorff and packing measures
11H60Mean value and transfer theorems