Cubic diophantine inequalities. (English) Zbl 0837.11016
H. Davenport and K. F. Roth [Mathematika, 2, 81–96 (1955; Zbl 0066.29301)] showed that for \(s \geq 8\) the values of the real additive form \(\lambda_1 x^3_1 + \ldots + \lambda_s x^3_s\) on \(\mathbb{Z}^s\) are dense on the real line, provided that \(\lambda_1/ \lambda_2\) is irrational. The authors prove (a quantitative form of) this result for \(s = 7\). The Fourier transform is used in essentially the same way as in Davenport and Roth to reduce the hardest part of the problem to giving good bounds for mean values of cubic exponential sums.
Recently T. D. Wooley [Invent. Math. 122, No. 3, 421–451 (1995; Zbl 0851.11055)] sharpened these mean value results for all real exponents \(s > 4\). The cases \(s = 6\) (discussed in Wooley’s paper) and \(s = 20/3\) (calculated here by an iterative procedure based on Wooley’s paper) are crucial here.
Another important role is played by R. C. Vaughan’s paper in [Acta Math. 162, 1–71 (1989; Zbl 0665.10033)]. A version of this which allows individual values, as well as mean values, to behave ‘well enough’ for the problem under discussion has been given by J. Brüdern, [Sémin. Théor. Nombres, Paris, 1990–1991, Prog. Math. 108, 23–34 (1993; Zbl 0815.11040)].
Recently T. D. Wooley [Invent. Math. 122, No. 3, 421–451 (1995; Zbl 0851.11055)] sharpened these mean value results for all real exponents \(s > 4\). The cases \(s = 6\) (discussed in Wooley’s paper) and \(s = 20/3\) (calculated here by an iterative procedure based on Wooley’s paper) are crucial here.
Another important role is played by R. C. Vaughan’s paper in [Acta Math. 162, 1–71 (1989; Zbl 0665.10033)]. A version of this which allows individual values, as well as mean values, to behave ‘well enough’ for the problem under discussion has been given by J. Brüdern, [Sémin. Théor. Nombres, Paris, 1990–1991, Prog. Math. 108, 23–34 (1993; Zbl 0815.11040)].
Reviewer: R.C.Baker (Provo)
References:
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| [5] | DOI: 10.1112/jlms/s2-39.2.205 · Zbl 0677.10034 · doi:10.1112/jlms/s2-39.2.205 |
| [6] | Davenport, Mathematika 3 pp 81– (1956) |
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| [12] | Vaughan, Ada Math. 162 pp 1– (1989) |
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