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On the cardinality of an exceptional set in a binary additive problem of the Goldbach type. (English) Zbl 1060.11062

Heath-Brown, D. R. (ed.) et al., Proceedings of the session in analytic number theory and Diophantine equations held in Bonn, Germany, January–June, 2002. Bonn: Univ. Bonn, Mathematisches Institut. Bonner Mathematische Schriften 360, 18 p. (2003).
Let \(\lambda_1\) and \(\lambda_2\) be positive real numbers, and write \(T_2(x)\) for the number of positive integers \(n\) up to \(x\) such that there exist no primes \(p_1\), \(p_2\) satisfying the equation \([\lambda_1p_1]+[\lambda_2p_2]=n\), where \([\lambda]\) denotes the integral part of \(\lambda\). Moreover, for given co-prime natural numbers \(d\) and \(l\), let \(T_3(x)\) be the number of \(n\leq x\) for which the above equation holds for no primes \(p_1\), \(p_2\) with \(p_1\equiv p_2\equiv l\pmod d\).
In this paper, the authors prove that if \(\lambda_1/\lambda_2\) is an algebraic irrational number, then one has \(T_2(x)\ll_{\varepsilon}x^{2/3+\varepsilon}\) for any \(\varepsilon>0\), substantially improving the previous result that Brüdern proved in 2000. The implicit constant in the result is ineffective, since Roth’s theorem is used in the proof. The main strategy of the proof is an investigation of the number of prime solutions \(p_1\) and \(p_2\) of the Diophantine inequality \(| \lambda_1p_1+[\lambda_2p_2]-n-1/2| <1/4\) by a method resembling the Hardy-Littlewood method.
It is also mentioned that one can show the same bound for \(T_3(x)\), that is, \(T_3(x)\ll_{\varepsilon}x^{2/3+\varepsilon}\) for any \(\varepsilon>0\), provided that \(d<x^{2-\varepsilon}\). It is pointed out, without a proof, that the latter theorem follows by the methods of this paper and of the second author [Vestn. Mosk. Univ., Ser. I 2000, No. 3, 57–61 (2000; Zbl 0986.11064) and “On two binary additive problems over prime numbers,” Proc. IV Int. Conf. “Modern problems of Number Theory and its applications.” Tula, Vol. 1, ChebyshevskiĭSb. 1, 72–94 (2001; Zbl 1213.11177)]. However, the reviewer is afraid that the statement of the latter theorem might contain a typographical error, because if \(d>(x+1)/\lambda_1\), then there is only at most one prime \(p_1\) such that \([\lambda_1p_1]\leq x\) and \(p_1\equiv l\pmod d\).
For the entire collection see [Zbl 1050.11003].

MSC:

11P32 Goldbach-type theorems; other additive questions involving primes
11D75 Diophantine inequalities
11D85 Representation problems
11P55 Applications of the Hardy-Littlewood method
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