Bounds for the cubic Weyl sum

Abstract

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 $$ \sum\limits_{n \leqslant N} {e\left( {\alpha {n^3}} \right){ \ll_{\varepsilon, \alpha }}{N^{\tfrac{5}{7} + \varepsilon }}} $$ for any $$ \varepsilon > 0 $$ and any quadratic irrational $$ \alpha \in \mathbb{R} - \mathbb{Q} $$ . Classically one would have had the (unconditional) exponent $$ \frac{3}{4} + \varepsilon $$ for such α. Bibliography: 5 titles.