Relations between exceptional sets for additive problems

Koichi Kawada; Trevor D. Wooley

doi:10.1112/jlms/jdq036

Relations between exceptional sets for additive problems

Koichi Kawada^*
, Trevor D. Wooley

^*Corresponding author for this work

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

19 Citations (Scopus)

Abstract

We describe a method for bounding the set of exceptional integers not represented by a given additive form in terms of the exceptional set corresponding to a subform. Illustrating our ideas with examples stemming from Waring's problem for cubes, we show, in particular, that the number of positive integers not exceeding N that fail to have a representation as the sum of six cubes of natural numbers is O(N(3/7)).

Original language	English
Pages (from-to)	437-458
Number of pages	22
Journal	Journal of the London Mathematical Society
Volume	82
DOIs	https://doi.org/10.1112/jlms/jdq036
Publication status	Published - Oct 2010

Keywords

WARINGS PROBLEM
HIGHER POWERS
SUMS
CUBES
IMPROVEMENTS
SQUARES
NUMBER

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title = "Relations between exceptional sets for additive problems",

abstract = "We describe a method for bounding the set of exceptional integers not represented by a given additive form in terms of the exceptional set corresponding to a subform. Illustrating our ideas with examples stemming from Waring's problem for cubes, we show, in particular, that the number of positive integers not exceeding N that fail to have a representation as the sum of six cubes of natural numbers is O(N(3/7)).",

keywords = "WARINGS PROBLEM, HIGHER POWERS, SUMS, CUBES, IMPROVEMENTS, SQUARES, NUMBER",

author = "Koichi Kawada and Wooley, \{Trevor D.\}",

year = "2010",

month = oct,

doi = "10.1112/jlms/jdq036",

language = "English",

volume = "82",

pages = "437--458",

journal = "Journal of the London Mathematical Society",

issn = "0024-6107",

publisher = "John Wiley \& Sons, Ltd.",

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T1 - Relations between exceptional sets for additive problems

AU - Kawada, Koichi

AU - Wooley, Trevor D.

PY - 2010/10

Y1 - 2010/10

N2 - We describe a method for bounding the set of exceptional integers not represented by a given additive form in terms of the exceptional set corresponding to a subform. Illustrating our ideas with examples stemming from Waring's problem for cubes, we show, in particular, that the number of positive integers not exceeding N that fail to have a representation as the sum of six cubes of natural numbers is O(N(3/7)).

AB - We describe a method for bounding the set of exceptional integers not represented by a given additive form in terms of the exceptional set corresponding to a subform. Illustrating our ideas with examples stemming from Waring's problem for cubes, we show, in particular, that the number of positive integers not exceeding N that fail to have a representation as the sum of six cubes of natural numbers is O(N(3/7)).

KW - WARINGS PROBLEM

KW - HIGHER POWERS

KW - SUMS

KW - CUBES

KW - IMPROVEMENTS

KW - SQUARES

KW - NUMBER

U2 - 10.1112/jlms/jdq036

DO - 10.1112/jlms/jdq036

M3 - Article (Academic Journal)

SN - 0024-6107

VL - 82

SP - 437

EP - 458

JO - Journal of the London Mathematical Society

JF - Journal of the London Mathematical Society

ER -

Relations between exceptional sets for additive problems

Abstract

Keywords

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