Almost equal summands in Waring\u2019s problem with shifts

Kirsti D Biggs

doi:10.1007/s00605-018-1178-7

Abstract

A result of Wright from 1937 shows that there are arbitrarily large natural numbers which cannot be represented as sums of skth powers of natural numbers which are constrained to lie within a narrow region. We show that the analogue of this result holds in the shifted version of Waring’s problem.

Highlights

This problem was originally studied by Chow in [3]
In [1], the author showed that an asymptotic formula for the number of solutions to (1) can be obtained whenever k ≥ 4 and s ≥ k2 + (3k − 1)/4
In 1937, Wright studied this question in the setting of the classical version of Waring’s problem, and proved in [6] that there exist arbitrarily large natural numbers n which cannot be represented as sums of s kth powers of natural numbers xi satisfying the condition xik − n/s < n1−1/2kφ(n) for 1 ≤ i ≤ s, no matter how large s is taken

Summary

Introduction

This problem was originally studied by Chow in [3]. In [1], the author showed that an asymptotic formula for the number of solutions to (1) can be obtained whenever k ≥ 4 and s ≥ k2 + (3k − 1)/4. Θs ∈ (0, 1) with θ1 ∈/ Q, we can find solutions in natural numbers xi to the following inequality, for all sufficiently large τ ∈ R: (x1 − θ1)k + · · · + (xs − θs )k − τ < η. In [1], the author showed that an asymptotic formula for the number of solutions to (1) can be obtained whenever k ≥ 4 and s ≥ k2 + (3k − 1)/4.

Results

Conclusion