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Diophantine approximation by primes. (English) Zbl 1257.11035
Let \(\lambda_1, \lambda_2, \lambda_3\) be non-zero real numbers, not all of the same sign and not all in rational ratio. The author uses the Davenport-Heilbronn method to show that for every real number \(\lambda_0\) and every \(\delta>0\) the Diophantine inequality \[ |\lambda_0+\lambda_1p_1+\lambda_2p_2+\lambda_3p_3| < (\max p_i)^{-2/9+\delta} \] has infinitely many solutions in primes \(p_1, p_2, p_3\). The approach dates to work of R. C. Vaughan [Proc. Lond. Math. Soc., III. Ser. 28, 373–384 (1974; Zbl 0274.10045)], who obtained the exponent \(1/10\) in place of \(2/9\), and incorporates ideas of G. Harman [J. Lond. Math. Soc., II. Ser. 44, No. 2, 218–226 (1991; Zbl 0754.11010)], who had obtained the best previously known exponent of \(1/5\). As expected, one must look for solutions in \((X,2X]^3\), where \(X=q^{3/2}\) and \(q\) is a denominator of a convergent to some \(\lambda_i/\lambda_j\). In fact, it is well-known that there exist coefficients \(\lambda_i\) and arbitrarily large \(X\) for which the inequality has no solution in \((X,2X]^3\). The author’s set-up uses the Brüdern-Fouvry vector sieve to obtain a lower bound for the characteristic function of pairs of primes in terms of upper and lower bounds \(\rho^{\pm}(n)\) for the characteristic function of primes. The functions \(\rho^{\pm}(n)\) are constructed by repeated application of Buchstab’s identity and numerical integration, in such a way that the mean square error in approximating the respective counting functions is small enough to handle the major arc. The author further argues that the exponential sums arising from these \(\rho^{\pm}(n)\) can be decomposed into suitable type I and type II sums, which form the basis of the minor arc analysis. The strategy is to bound the measure of the set on which all three generating functions are nearly as large as \(X^{7/9+\delta}\), and this is the heart of the paper. Large exponential sums \(S(\lambda_i\alpha)\) produce good rational approximations \(a_i/q_i\) to \(\lambda_i\alpha\), and upper bounds for averages of \(S(a_i/q_i+\beta_i)\) over \(q_i\) then yield upper bounds on the number of such \(a_i/q_i\) that can occur. For large moduli the type II average estimates are obtained via the large sieve, and for small moduli they are obtained from classical mean value estimates for Dirichlet polynomials.

MSC:
11D75 Diophantine inequalities
11P32 Goldbach-type theorems; other additive questions involving primes
11N36 Applications of sieve methods
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