The Davenport-Heilbronn Fourier transform method, and some Diophantine inequalities (Q2707582)

The device of ``pruning the major arcs'' in the circle method (for which see, for example, the author's paper [Math. Proc. Camb. Philos. Soc. 103, 27-33 (1988; Zbl 0655.10041)]) is extended to the Davenport-Heilbronn method for Diophantine inequalities, by a procedure involving the application of the Poisson sum formula to the Fourier transforms that arise naturally in the Davenport-Heilbronn method. NEWLINENEWLINENEWLINEThe power of this device is illustrated by the improvement of several known results, by a treatment that is possibly more straightforward than the previous ones. For example, the author shows that the values at integer points of the polynomial NEWLINE\[NEWLINE \lambda_1 x^2 + \lambda_2 y^2 + \lambda_3 z_1^3 + \lambda_4 z_2^3 + \lambda_5 z_3^a +\lambda_6 z_4^b ,NEWLINE\]NEWLINE where \(\lambda_i\) is real and the ratios \(\lambda_i/\lambda_j\) are not all rational, are dense on the real line provided only that one assumes the natural condition \(a^{-1}+b^{-1}> {1 \over 3}\) that also arises in the treatment by \textit{C.~Hooley} [Recent progress in analytic number theory, Symp. Durham 1979, Vol.~1, 127-191 (1981; Zbl 0463.10037)] of the corresponding equation relating to the representation of a large number~\(n\). For comparison, \textit{R.~C.~Baker} and \textit{G.~Harman} [J. Number Theory 18, 69-85 (1984; Zbl 0533.10014)] used the traditional Davenport-Heilbronn method and found they needed a more restrictive condition on \(a\) and~\(b\), and also needed to assume that \(\lambda_1/\lambda_2\) is irrational. NEWLINENEWLINENEWLINESimilar results are derived for the polynomials NEWLINE\[NEWLINE \lambda_1 x_1^2 +\sum_{k=2}^6 \lambda_k x_k^3,\qquad \sum_{k=1}^{16} \lambda_{k+1} x_k^{k+1}. NEWLINE\]NEWLINE Of these, the first corresponds to a theorem of G.~L.~Watson, more recently treated using ``pruning'' in the circle method by \textit{R.~C.~Vaughan} [Q. J. Math., Oxf. II Ser. 37, 117-127 (1986; Zbl 0589.10047)], on the representation of a number as the sum of one square and five cubes. The second corresponds to a problem most recently treated by \textit{K.~B.~Ford} [J. Am. Math. Soc. 9, 919-940 (1996; Zbl 0866.11054)], who represented each large number, using fewer variables, as \(\sum_{1 \leq k \leq 14}x_k^{k+1}\). The larger number of variables in the author's theorem arises from an additional appeal to Weyl's inequality that appears to be essential.NEWLINENEWLINEFor the entire collection see [Zbl 0932.00040].