Multiplicative zero-one laws and metric number theory. (English) Zbl 1292.11085

Beresnevich, Victor; Haynes, Alan; Velani, Sanju

Acta Arith. 160, No. 2, 101-114 (2013).

The authors study multiplicative analogues of the celebrated Duffin–Schaeffer conjecture [R. J. Duffin and A. C. Schaeffer, Duke Math. J. 8, 243–255 (1941; Zbl 0025.11002)] in metric diophantine approximation. To be precise, for a function \(\psi: {\mathbb N} \rightarrow [0,1/2)\), they study the set \({\mathcal S}^\times_n (\psi)\) consisting of the points \((x_1, \dots, x_n) \in [0,1]^n\) for which the inequality \[ \prod_{i=1}^n \| q x_i \| < \psi(q) \] has infinitely many solutions \(q \in {\mathbb N}\), where \(\| \cdot \|\) denotes the distance to the nearest integer. Additionally, they study the set \({\mathcal D}^\times_n (\psi)\), in which the \(\| qx_i \|\) is replaced with the distance to the nearest integer co-prime to \(q\).
It is shown that in both cases, the sets satisfy a zero-one law, i.e. the Lebesgue measure of either set must be equal to zero or one. Additionally, a criterion under which the measure must be equal to one is derived in the spirit of the Duffin–Schaeffer theorem [loc. cit.]. The conditions are fairly technical, and the authors state conjectures giving a simpler necessary and sufficient conditions for the measure to be equal to one. Finally, the second main result is applied to obtain a metrical result related to the \(p\)-adic Littlewood conjecture. At the heart of the proofs is a ‘cross-fibering technique’, allowing one to reverse the implications in Fubini’s theorem.

Reviewer: Simon Kristensen (Aarhus)

MSC:

11J83	Metric theory
11J13	Simultaneous homogeneous approximation, linear forms
11K60	Diophantine approximation in probabilistic number theory

Keywords:

zero-one law; metric multiplicative diophantine approximation; Duffin–Schaeffer theorem

Citations:

Zbl 0025.11002