Geodesic
Bounds for the cubic Weyl sum
Zapiski Nauchnykh Seminarov POMI, Studies in number theory. Part 10, Tome 377 (2010), pp. 199-216 Cet article a éte moissonné depuis la source Math-Net.Ru

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Subject to the $abc$-conjecture, we improve the standard Weyl estimate for cubic exponential sums in which the argument is a quadratic irrational. Specifically we show that $$ \sum_{n\le N}e(\alpha n^3)\ll_{\varepsilon,\alpha}N^{\frac57+\varepsilon} $$ for any $\varepsilon>0$ and any quadratic irrational $\alpha\in\mathbb R-\mathbb Q$. Classically one would have had the (unconditional) exponent $\frac34+\varepsilon$ for such $\alpha$. Bibl. 5 titles. 
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     author = {D. R. Heath-Brown},
     title = {Bounds for the cubic {Weyl} sum},
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     url = {http://geodesic.mathdoc.fr/item/ZNSL_2010_377_a15/}
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D. R. Heath-Brown. Bounds for the cubic Weyl sum. Zapiski Nauchnykh Seminarov POMI, Studies in number theory. Part 10, Tome 377 (2010), pp. 199-216. http://geodesic.mathdoc.fr/item/ZNSL_2010_377_a15/

[1] E. Bombieri, S. Sperber, “On the estimation of certain exponential sums”, Acta Arith., 69 (1995), 329–358 | MR | Zbl

[2] J. H. Loxton, R. C. Vaughan, “The estimation of complete exponential sums”, Canad. Math. Bull., 28 (1985), 440–454 | DOI | MR | Zbl

[3] C. J. Ringrose, The $q$-analogue of van der Corput's method, DPhil Thesis, Oxford, 1985

[4] R. C. Vaughan, “Some remarks on Weyl sums”, Topics in classical number theory (Budapest, 1981), v. I, II, Colloq. Math. Soc. János Bolyai, 34, North-Holland, Amsterdam, 1984, 1585–1602 | MR

[5] H. Weyl, “Über die Gleichverteilung der Zahlen mod. Eins”, Math. Ann., 77 (1916), 313–352 | DOI | MR | Zbl