The Quartic Weyl Sum

11:00 am, Monday, 27th November

Abstract

The k-th power Weyl sum is S_k(N,a)=\sum_{n\le N} \exp(2\pi i an^k), where a is a real parameter. The classical bound takes the form O_{k,a,c}(N^c), for any c > 1-2^{1-k}, whenever a is well-approximable by rationals. This is best possible for k=2, and has not been improved for 100 years for k=3 or k=4. The talk will show how to replace the condition c > 7/8 for k=4 by c > 5/6, whenever a is a quadratic irrarional.