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

Jörg Brüdern; Trevor D. Wooley

doi:10.1007/s00209-016-1759-x

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

Jörg Brüdern^*, Trevor D. Wooley

^*Corresponding author for this work

Research output: Contribution to journal › Article (Academic Journal) › peer-review

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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≥4.

Original language	English
Pages (from-to)	179-196
Number of pages	18
Journal	Mathematische Zeitschrift
Volume	286
Issue number	1-2
Early online date	13 Oct 2016
DOIs	https://doi.org/10.1007/s00209-016-1759-x
Publication status	Published - 8 May 2017

Keywords

Sums of cubes
sums of k-th powers
Hardy-Littlewood method

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http://arxiv.org/abs/1506.01902

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abstract = "We establish that, for almost all natural numbers N, there is a sum of two positive integral cubes lying in the interval [N−N7/18+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≥4. ",

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AU - Wooley, Trevor D.

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N2 - We establish that, for almost all natural numbers N, there is a sum of two positive integral cubes lying in the interval [N−N7/18+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≥4.

AB - We establish that, for almost all natural numbers N, there is a sum of two positive integral cubes lying in the interval [N−N7/18+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≥4.

KW - Sums of cubes

KW - sums of k-th powers

KW - Hardy-Littlewood method

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JO - Mathematische Zeitschrift

JF - Mathematische Zeitschrift

IS - 1-2

ER -

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

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