A Diophantine problem with a prime and three squares of primes (Q455808)

Let \(\lambda_1, \dots, \lambda_4\) be nonzero real numbers, not all of the same sign, with \(\lambda_1/\lambda_2 \not \in {\mathbb Q}\). The authors show that for every real number \(\varpi\) and every \(\varepsilon > 0\) the inequality NEWLINE\[NEWLINE|\lambda_1 p_1 + \lambda_2 p_2^2 + \lambda_3 p_3^2 + \lambda_4 p_4^2 + \varpi| < (\max p_j)^{-1/18+\varepsilon}NEWLINE\]NEWLINE has infinitely many solutions in primes \(p_1, \dots, p_4\). This improves on earlier work of \textit{W. Li} and \textit{T. Wang} [Ramanujan J. 25, No. 3, 343--357 (2011; Zbl 1234.11036)], in which the exponent 1/18 was replaced by 1/28. The proof uses the Davenport-Heilbronn version of the circle method, and the new savings arise from a widening of the major arc, as in previous work of the first author and \textit{V. Settimi} [Acta Arith. 154, No. 4, 385--412 (2012; Zbl 1306.11031)]. The minor (or intermediate) arcs are handled using an estimate of \textit{A. Ghosh} [Proc. Lond. Math. Soc. (3) 42, 252--269 (1981; Zbl 0397.10026)] for the exponential sum over squares of primes, together with a well-known bound of R.\ C.\ Vaughan for sums over primes. The authors observe that this refinement can also be applied to improve their earlier result [Acta Arith. 145, No. 2, 193--208 (2010; Zbl 1222.11049)] on the real analogue of the Linnik-Goldbach problem involving two primes and a bounded number of powers of two.