On the measure of large trigonometric sums

trigs.dvi.gz (68k)

Abstract: We give an almost complete solution to the following question: For fixed $k$ and $u$, what is the sum $p(z)$ of $k$ distinct integral powers of the complex variable $z$ such that the set of $z$ values on the unit circle where $p(z)$ has absolute value greater than $k-u$ has maximal measure. It turns out that when $u$ is sufficiently small compared to $k$ most of the exponents form an arithmetical progression and we also describe the exponents not in this progression.