Wilson, Kathryn2025-04-152025-04-152025https://hdl.handle.net/2097/44931This 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 . . . See PDF file for full abstract.en-USWaring numberDiophantine inequalitiesHardy-Littlewood circle methodFunction fieldsCongruencesDavenport-Heilbronn methodDissertation