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Diophantine approximation by prime numbers. (English) Zbl 0754.11010
The author proves: Suppose that \(\lambda_ 1,\lambda_ 2,\lambda_ 3\) are non-zero real numbers not all of the same sign, that \(\alpha\) is real and that \(\lambda_ 1/\lambda_ 2\) is irrational. Then, for any \(\varepsilon>0\), there are infinitely many ordered triples of primes \(p_ 1,p_ 2,p_ 3\) for which \[ |\alpha+\lambda_ 1p_ 1+\lambda_ 2p_ 2+\lambda_ 3p_ 3|<(\max p_ j)^{- (1/5)+\varepsilon}. \] Results of this type were first considered by W. Schwarz [J. Reine Angew. Math. 212, 150-157 (1963; Zbl 0116.036)] and I. Danicic [Can. J. Math. 18, 621-628 (1966; Zbl 0224.10022)]. The latest known result, with the exponent 1/5 replaced by 1/6, is due to R. C. Baker and the author [J. Lond. Math. Soc., II. Ser. 25, 201- 215 (1982; Zbl 0443.10015)]. The improvement is obtained by considering suitable averages of exponential sums over primes.

MSC:
11D75 Diophantine inequalities
11J25 Diophantine inequalities
11L03 Trigonometric and exponential sums (general theory)
11L20 Sums over primes
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