Bounds for the cubic Weyl sum

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.