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 Γ∗(k,pn) be the smallest positive integer s such that for any integers ai coprime to p and integer a, the congruence a1x1k+⋯+asxsk≡a(modpn)is solvable in integers xi, with p∤xi, 1≤i≤s. We prove that for ε>0, $$\Gamma^*(k,p^n) \ll_\varepsilon k^{\frac 1{\phi(t_1)}+\varepsilon},$$ where t1 is the number of nonzero k-th powers mod p. The estimate is best possible aside from the possible removal of the ε. Next, using a result of Chow, Lindqvist and Prendiville \cite{CLP21}, we establish that for any positive integer k≥4, 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 ai, with p∤ai for 1≤i≤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≤xi≪kp1k for 1≤i≤s. Lastly, we will discuss how the Bentkus--Götze--Freeman variant of the Davenport--Heilbronn circle method can be used to study Fq[t] solutions to inequalities of the form [\mathrm{ord}(\lambda_1x_1^k+\cdots+\lambda_sx_s^k-\gamma)<\tau] where constants λ1,…,λs∈Fq((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.]