On exceptional sets for numbers representable by binary sums (Q1359144)

This paper contains several results concerning numbers representable as binary sums of the type \(a_1p+a_2n^k\), \(a_1p_1+a_2p_2^k\) and \(a_1p+a_2P(n)\), where \(k\geq2\) is an integer, \(P\) is a polynomial of degree 2 with leading coefficient 1, and \(p\) denotes a prime number. The authors prove that all sufficiently large integers satisfying a necessary arithmetical condition have at least one such representation, with an exceptional set of at most \(X^{\theta}\) integers \(n\leq X\), where \(\theta=\theta(k)<1\) does not depend on \(a_1\), \(a_2\) or the coefficients of \(P\). Here, ``sufficiently large'' is explicitly given in terms of these coefficients. Their results extend and include previous results of \textit{R. Brünner, A. Perelli} and \textit{J. Pintz} [Acta Math. Hung. 53, 347-365 (1989; Zbl 0683.10039)], \textit{W. Schwarz} [Math. Nachr. 23, 327-348 (1961; Zbl 0103.27502)], \textit{V. A. Plaksin} [Math. Notes 47, 278-286 (1990); translation from Mat. Zametki 47, 78-90 (1990; Zbl 0708.11053)], \textit{A. I. Vinogradov} [Acta Arith. 46, 33-56 (1985; Zbl 0597.10048)] and \textit{A. Zaccagnini} [Mathematika 39, 400-421 (1992; Zbl 0760.11026)], concerning conjectures H and L of \textit{G. H. Hardy} and \textit{J. E. Littlewood} [Acta Math. 44, 1-70 (1923; JFM 48.0143.04)]. The main novelties lie in the good uniform bounds in terms of the coefficients, and the significant simplifications over the previous proofs of the above results, which are due to a more efficient implementation of the variant of the circle method introduced by \textit{H. L. Montgomery} and \textit{R. C. Vaughan} [Acta Arith. 27, 353-370 (1975; Zbl 0478.10033)], already exploited by \textit{M.-C. Liu} and \textit{K.-M. Tsang} in [Théorie des nombres, C. R. Conf. Int., Quebec 1987, 595-624 (1989; Zbl 0682.10043)].