Approximations to Weyl sums
by
Jörg Brüdern
Universität Stuttgart
Coauthors: Dirk Daemen

In this talk, we discuss the error between a Weyl sum ∑_{n ≤ N} e(an)^k and its standard approximation q^-1 S(q, a) v(a-a/q) where S(q, a) = ∑_x=1^q e(ax^k/q) and v(b)=∫₀^N e(bt^k)dt. This error is known to be O(q^1/2+e(1+N^k|a-a/q|)^1/2, but it has been suggested that the error is actually much smaller, and that perhaps the exponents 1/2 can be reduced to 1/k. We shall show that this is not the case: the known estimate is essentially optimal, pointwise and in various quadratic means.

Date received: May 9, 2008