Weyl's Equidistribution Theorem in function fields and multivariable generalizations

Abstract

This thesis is concerned with the problem of finding a satisfactory function field analogue to Weyl's Equidistribution Theorem, a task that was initiated by Carlitz in 1952. More specifically, we are looking at the equidistribution of polynomial values $f(x)$ as $x$ runs over the ring $\mathbb{F}_q[T]$, where $f(X)$ is a polynomial with coefficients taken from $\mathbb{F}_q\big((T^{-1})\big)$, the field of formal Laurent series in $T^{-1}$. Classically, results of this type were restricted to the case where the degree of $f$ is less than $p:=\text{char}\,\mathbb{F}_q$, and this \textit{characteristic barrier} was broken about a decade ago by L\^e, Liu and Wooley using new developments surrounding Vinogradov's Mean Value Theorem.

Here, we resolve a remaining conjecture made by L\^e, Liu and Wooley, thus establishing the largest class of equidistributed polynomial sequences $f(x)$ determined by irrationality conditions on the coefficients of $f(X)$. We also consider further generalisations of the resulting theorem, which we phrase in terms of additive polynomials related to $f(X).$ In any case, the main difficulty that we encounter arise from some possible interference occurring between terms of the form $\alpha X^k$ with those of the form $\beta X^{p^\upsilon k}$ appearing in the expansion of $f(X)$. To avoid problems of this type, we introduce a transformation $f(X)\mapsto f^\tau(X)$ which preserve the size of Weyl sums, and has the property that $f^\tau(X)$ does not involve any terms of the form $\beta X^{pk}$.

In a different but related direction, we generalize the method of L\^e, Liu and Wooley for multivariate polynomial sequences $f(x_1,...,x_d)$ where $(x_1,...,x_d)$ run over $\mathbb{F}_q[T]^d$, and we also consider the case where each of $x_1,...,x_d$ is required to be monic. Similarly to the original paper, the method consists in establishing a minor arc estimate for multivariate Weyl sums using the Large Sieve Inequality together with a multivariate version of Vinogradov's Mean Value Theorem in function fields obtained previously by Kuo, Liu and Zhao.