Counting solutions to diagonal congruences and Diophantine inequalities

Abstract

This dissertation will cover three results. Each result has a connection to Waring's problem and the circle method. Throughout, let $p$ be prime, and let $n$, $k$ and $s$ be positive integers.

First, let $\Gamma^*(k,p^n)$ be the smallest positive integer $s$ such that for any integers $a_i$ coprime to $p$ and integer $a$, the congruence $$ a_1x_1^k + \cdots + a_sx_s^k \equiv a \pmod {p^n} $$ is solvable in integers $x_i$, with $p \nmid x_i$, $1 \le i \le s$. We prove that for $\varepsilon>0$, $$\Gamma^*(k,p^n) \ll_\varepsilon k^{\frac 1{\phi(t_1)}+\varepsilon},$$ where $t_1$ is the number of nonzero $k$-th powers mod $p$. The estimate is best possible aside from the possible removal of the $\varepsilon$.

Next, using a result of Chow, Lindqvist and Prendiville \cite{CLP21}, we establish that for any positive integer $k\geq4$, positive integer $s$ such that $$s\geq\frac{3}{2}k(\log\,k+\log\,\log\,k+2+O(\log\,\log\,k/\log\,k)),$$ prime $p$, integer $a$, and integers $a_i$, with $p\nmid a_i$ for $1\leq i\leq s$, there exists a solution of \[a_1x_1^k+a_2x_2^k+\cdots+a_sx_s^k\equiv a\qquad(\text{mod }p) \] with $1\leq x_i\ll_{k}p^{\frac{1}{k}}$ for $1\leq i\leq s$.

Lastly, we will discuss how the Bentkus--Götze--Freeman variant of the Davenport--Heilbronn circle method can be used to study $\mathbb{F}_q[t]$ solutions to inequalities of the form \[\mathrm{ord}(\lambda_1x_1^k+\cdots+\lambda_sx_s^k-\gamma)<\tau\] where constants $\lambda_1,\dots, \lambda_s \in\mathbb{F}_q((1/t))$ satisfy certain conditions. This result is a generalization of the work done by Spencer in \cite{CVS} to count the number of solutions to inequalities of the form \[\mathrm{ord}(\lambda_1x_1^k+\cdots+\lambda_sx_s^k)<\tau.\]