On a Diophantine problem with one prime, two squares of primes and \(s\) powers of two (Q2895960)

Let \(\lambda_i\) and \(\mu_i\) be nonzero real numbers with \(\lambda_1 < 0\), \(\lambda_2, \lambda_3 > 0\), \(\lambda_2/\lambda_3 \not \in {\mathbb Q}\), and \(\lambda_i/\mu_i \in {\mathbb Q} \;(i=1,2,3)\). The authors show that for all real numbers \(\eta > 0\) and \(\varpi\), there exists an integer \(s_0\) such that, whenever \(s \geqslant s_0\), the inequality NEWLINE\[NEWLINE|\lambda_1 p_1 + \lambda_2 p_2^2 + \lambda_3 p_3^2 + \mu_1 2^{m_1} + \dots + \mu_s 2^{m_s} + \varpi| < \etaNEWLINE\]NEWLINE has infinitely many solutions in primes \(p_1, p_2, p_3\) and positive integers \(m_1, \dots, m_s\). Here the admissible value of \(s_0\) is given explicitly in terms of the \(\lambda_i\) and the denominators of the \(\lambda_i/\mu_i\). There is a long history of work on related questions arising as variants of the Waring-Goldbach and Goldbach-Linnik problems, and this particular result improves on work of \textit{W. Li} and \textit{T. Wang} [Pure Appl. Math. 21, No. 4, 295--299 (2005; Zbl 1183.11018)]. The proof is accomplished via the Davenport-Heilbronn version of the circle method, with the improvement stemming from an enlarged major arc on which an \(L^2\)-estimate for the exponential sum over prime squares can be applied. The narrower minor arcs are then handled using an estimate of \textit{A. Ghosh} [Proc. Lond. Math. Soc. (3) 42, 252--269 (1981; Zbl 0397.10026)] for the sum over prime squares, the techniques of the first author and \textit{A. Zaccagnini} [Acta Arith. 145, No. 2, 193--208 (2010; Zbl 1222.11049)] for the sum over primes, and the algorithm of \textit{J. Pintz} and \textit{I. Z. Ruzsa} [Acta Arith. 109, No. 2, 169--194 (2003; Zbl 1031.11060)] for the sum over powers of two.