Jérémy Champagne, University of Waterloo
Weyl's Equidistribution Theorem in function fields and multivariable generalizations
This thesis is concerned with finding a suitable function field analogue to the classical equidistribution theorem of Weyl. More specifically, we are interested in the distribution of polynomial values f(x) as x runs over the ring Fq[T], and where the coefficients of f(X) are taken from the field of formal power series Fq((1/T)). Classically, results of this type were all subject to the constraint deg f <p where p:=char(Fq). In 2013, Lê, Liu and Wooley were able to break this characteristic barrier using modern developments regarding Vinogradov's Mean Value Theorem.
The first set of results in this thesis consists in a resolution of the main conjecture made by Lê-Liu-Wooley, which establishes the largest possible class of equidistributed polynomial sequences f(x) that can be determined by irrationality conditions on the coefficients of f(X). This is done by introducing a new transformation f(X) -> f^τ(X) which preserves the size of Weyl sums, and is such that f^τ(X) does not involve any powers divisible by p.
The second sets of results is concerned with a multivariate generalization of the method of Lê-Liu-Wooley. As such, we use a multivariate version of Vinogradov's Mean Value Theorem together with the Large Sieve Inequality to obtain suitable minor arc estimates for Weyl sums in d variables. We then use these minor arc estimates to study the distribution of polynomial values f(x_1,...,x_d)$ as (x_1,...,x_d) runs over Fq[T]^d, and we also consider the case where each of x_1,...,x_d is required to be monic.
MC 6029