Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Partitio Numerorum: sums of squares and higher powers

Joerg Bruedern, Trevor D. Wooley

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 $k$-th powers is at least $\lfloor c_0k\rfloor +2$, where $c_0=2.136294\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.

Partitio Numerorum: sums of squares and higher powers

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 -th powers. We show that such representations exist when the number of -th powers is at least , where . 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.
Paper Structure (9 sections, 33 theorems, 226 equations)

This paper contains 9 sections, 33 theorems, 226 equations.

Table of Contents

  1. Introduction
  2. A certain counting problem
  3. The circle method: initial steps
  4. Minor arcs: pruning by height
  5. Enhancements: pruning by size
  6. Pruning by size for squares
  7. The role of Hölder's inequality
  8. Outlook
  9. Appendix: An exponential sum

Key Result

Theorem 1.1

Let $k\geqslant 3$. Then where $c_0={\frac{3}{4}}+2\log 2=2.136294\ldots$. Moreover, one has

Theorems & Definitions (51)

  • Theorem 1.1
  • Lemma 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Theorem 2.5
  • ...and 41 more