Geodesic
Bounds for the cubic Weyl sum
Zapiski Nauchnykh Seminarov POMI, Studies in number theory. Part 10, Tome 377 (2010), pp. 199-216

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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. 
@article{ZNSL_2010_377_a15,
     author = {D. R. Heath-Brown},
     title = {Bounds for the cubic {Weyl} sum},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {199--216},
     publisher = {mathdoc},
     volume = {377},
     year = {2010},
     language = {en},
     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/