×

The \(k\)-dimensional Duffin and Schaeffer conjecture. (English) Zbl 0714.11048

The following \(k\)-dimensional analogue of the Duffin and Schaeffer conjecture is proved: Let \(k>1\) and let \(\{\alpha_ n\}\) denote a sequence of real numbers with \(0\leq \alpha_ n<1/2\), and suppose that the series \(\sum^{\infty}_{n=1}(\alpha_ n\phi (n)/n)^ k\) diverges. Then the inequalities \(\max (| x_ 1n-a_ 1|,...,| x_ kn-a_ k|)<\alpha_ n,\quad (a_ i,n)=1\quad (i=1,...,k)\) have infinitely many solutions for almost all \(x\in {\mathbb R}^ k\).
Reviewer: V.Ennola

MSC:

11K60 Diophantine approximation in probabilistic number theory
11J83 Metric theory
11J25 Diophantine inequalities
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML

References:

[1] Duffin, R.J. and Schaeffer, A.C., Khintchine’s problem in metric Diophantine approximation, Duke Math. J.8 (1941), 243-255. · JFM 67.0145.03
[2] Erdös, P., On the distribution of convergents of almost all real numbers, J. Number Theory2 (1970), 425-441. · Zbl 0205.34902
[3] Gallagher, P.X., Approximation by reduced fractions, J. Math. Soc. of Japan13 (1961), 342-345. · Zbl 0106.04106
[4] Halberstam, Richert, “Sieve methods,” Academic Press, London, 1974. · Zbl 0298.10026
[5] Sprindzuk, V.G., “Metric theory of Diophantine approximations,” V.H. Winston and Sons, Washington D.C., 1979. · Zbl 0482.10047
[6] Vaaler, J.D., On the metric theory of Diophantine approximation, Pacific J. Math.76 (1978), 527-539. · Zbl 0352.10026
[7] Vilchinski, V.T., On simultaneous approximations, Vesti Akad Navuk BSSR Ser Fiz.-Mat (1981), 41-47. · Zbl 0464.10040
[8] , The Duffin and Schaeffer conjecture and simultaneous approximations, Dokl. Akad. Nauk BSSR25 (1981), 780-783. · Zbl 0473.10034
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.