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Diophantine approximation by cubes of primes and an almost prime. II. (English) Zbl 1003.11046

Let \(\lambda_1,\ldots,\lambda_4\) be non-zero real numbers with \(\lambda_1/\lambda_2\) irrational and negative. Let \({\mathcal E}(N)={\mathcal E}(N,\delta)\) denote the Lebesgue measure of the set of real numbers \(\eta\) such that \(|\eta|\leq N\) and the inequality \[ |\lambda_1x^3+\lambda_2p_1^3+ \lambda_3p_2^3+\lambda_4p_3^3 -\eta|< N^{-\delta} \] has no solutions in primes \(p_1\), \(p_2\), \(p_3\) and a \(P_3\)-number \(x\) (a natural number \(x\) is called a \(P_r\)-number if the number of prime factors of \(x\) is at most \(r\), counted according to multiplicity). Then, the main theorem of this paper asserts that for some positive absolute constant \(\delta\), there exist arbitrarily large values of \(N\) for which one has \({\mathcal E}(N)\ll N\exp(-(\log N)^{1/4})\). The last estimate for \({\mathcal E}(N)\) is proved for all \(N\) in the case where \(\lambda_1/\lambda_2\) is algebraic, in addition to the above conditions.
This theorem improves the previous result of the second author [Rocky Mt. J. Math. 30, 961-980 (2000; Zbl 0973.11070)], where \(P_6\) was adopted in place of \(P_3\) above. A key to this improvement is an estimate for the number, say \(S\), of solutions of the Diophantine inequality \[ |\lambda(x_1^3-x_2^3)+\mu(y_1^3+y_2^3-y_3^3-y_4^3)|<1/2, \] subject to \(X<x_i\leq 2X\) and \(Y<y_j\leq 2Y\), where \(\lambda\) and \(\mu\) are fixed non-zero real numbers, \(X\geq 1\) and \(Y=X^{5/6}\). Actually, by an analogue of Vaughan’s diminishing range method in Waring’s problem, the authors show that \(S\ll X^{1+\varepsilon}Y^2\) for each fixed \(\varepsilon>0\).
They also point out that the following conclusion is derived from the main theorem via a kind of pigeonhole principle: If \(\lambda_1,\ldots,\lambda_8\) are non-zero real numbers such that \(\lambda_1/\lambda_2\) is irrational and negative, then the set of values taken by \(\lambda_1x^3+\lambda_2p_1^3+\cdots+ \lambda_8p_7^3\) with \(P_3\)-number \(x\) and primes \(p_i\) is dense in the set of all real numbers.

MSC:

11P32 Goldbach-type theorems; other additive questions involving primes
11P55 Applications of the Hardy-Littlewood method
11N36 Applications of sieve methods

Citations:

Zbl 0973.11070
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