September 2024 Partitio Numerorum: sums of squares and higher powers
Jörg Brüdern, Trevor D. Wooley
Funct. Approx. Comment. Math. 71(1): 21-67 (September 2024). DOI: 10.7169/facm/2165

Abstract

We survey the potential for progress in additive number theory arising from recent advances concerning major arc bounds associated with mean value estimates for smooth Weyl sums. We focus attention on the problem of representing large positive integers as sums of a square and a number of $k$-th powers. We show that such representations exist when the number of\break $k$-th powers is at least $\lfloor c_0 k\rfloor +2$, where $c_0=2.13629\ldots$. By developing an abstract framework capable of handling sequences with appropriate distribution properties, analogous conclusions are obtained, for example, when the square is restricted to have prime argument.

Citation

Download Citation

Jörg Brüdern. Trevor D. Wooley. "Partitio Numerorum: sums of squares and higher powers." Funct. Approx. Comment. Math. 71 (1) 21 - 67, September 2024. https://doi.org/10.7169/facm/2165

Information

Published: September 2024
First available in Project Euclid: 22 May 2024

Digital Object Identifier: 10.7169/facm/2165

Subjects:
Primary: 11P05 , 11P55

Keywords: Hardy-Littlewood method , Waring's Problem

Rights: Copyright © 2024 Adam Mickiewicz University

Vol.71 • No. 1 • September 2024
Back to Top