We establish an essentially optimal estimate for the ninth moment of the exponential sum having argument αx + βx. The first substantial advance in this topic for over 60 years, this leads to improvements in Heath-Brown’s variant of Weyl’s inequality, and other applications of Diophantine type.

A detailed discussion of the point in polygon problem for arbitrary polygons is given. Two concepts for solving this problem are known in literature: the even–odd rule and the winding number, the former leading to ray-crossing, the latter to angle summation algorithms. First we show by mathematical means that both concepts are very closely related, thereby developing a first version of an algor...