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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
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References:

[1] Duffin, R.J. and Schaeffer, A.C., Khintchine’s problem in metric Diophantine approximation, Duke Math. J.8 (1941), 243-255.
[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
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