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Additive representation in short intervals, II: sums of two like powers

Joerg Bruedern, Trevor D. Wooley

Abstract

We establish that, for almost all natural numbers $N$, there is a sum of two positive integral cubes lying in the interval $[N-N^{7/18+ε},N]$. Here, the exponent $7/18$ lies half way between the trivial exponent $4/9$ stemming from the greedy algorithm, and the exponent $1/3$ constrained by the number of integers not exceeding $X$ that can be represented as the sum of two positive integral cubes. We also provide analogous conclusions for sums of two positive integral $k$-th powers when $k\ge 4$.

Additive representation in short intervals, II: sums of two like powers

Abstract

We establish that, for almost all natural numbers , there is a sum of two positive integral cubes lying in the interval . Here, the exponent lies half way between the trivial exponent stemming from the greedy algorithm, and the exponent constrained by the number of integers not exceeding that can be represented as the sum of two positive integral cubes. We also provide analogous conclusions for sums of two positive integral -th powers when .

Paper Structure

This paper contains 6 sections, 14 theorems, 136 equations.

Table of Contents

  1. Introduction
  2. Infrastructure
  3. The collapse of the major arcs
  4. Minor arcs with a difference
  5. The annihilation of the central intervals
  6. Deduction of the main results

Key Result

Theorem \oldthetheorem

Suppose that $k\geqslant 3$. Then, whenever $Z\geqslant N^{{\theta}_k}$, one has

Theorems & Definitions (28)

  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • proof
  • ...and 18 more