Applications of the Hardy-Littlewood method to polynomial congruences and Diophantine inequalities

Date

2025

Journal Title

Journal ISSN

Volume Title

Publisher

Abstract

This manuscript revolves around two peer-reviewed results. First, we prove that for any positive integers k, q, n with n>N(k), integer c, and polynomials fi(x) of degree k whose leading coefficients are relatively prime to q, there exists a solution x to the congruence

i=1nfi(xi)≡c(modq)

that lies in a cube of side length at least max{q1/k,k}. Moreover, the result is best possible up to the determination of N(k). The latter half of the manuscript is centred around Diophantine inequalities. Let k≥2, s≥⌈k(logk+4.20032)⌉, and λ1,…,λs,ωR. Assume that the λi are non-zero, not all in rational ratio, and not all of the same sign in the case that k is even. Then, for any $\epsilon > 0 $, the inequality

|λ1x1k+λ2x2k+⋯+λsxsk+ω|<ϵ$$has$Psk$integersolutionswith$|xi|≤P$.Moreovertheasymptoticformulaforthenumberofsmoothsolutionsisestablishedassumingthesameconditionshold.

Description

Keywords

Davenport-Heilbronn method, circle method, exponential sums, Diophantine Inequalities, Congruence modulo q

Graduation Month

August

Degree

Doctor of Philosophy

Department

Department of Mathematics

Major Professor

Craig Spencer

Date

Type

Dissertation

Citation