On additive representation functions Article Swipe

View

YOU? · · 2018 · Open Access · · DOI: https://doi.org/10.5486/pmd.2018.8175

Let $A$ be an infinite set of natural numbers. For $n\in \mathbb{N}$, let $r(A, n)$ denote the number of solutions of the equation $n=a+b$ with $a, b\in A, a\le b$. Let $|A(x)|$ be the number of integers in $A$ which are less than or equal to $x$. In this paper, we prove that, if $r(A, n)\not= 1$ for all sufficiently large integers $n$, then $|A(x)|> \frac 12 (\log x/\log\log x)^2$ for all sufficiently large $x$.

Related Topics

Concepts

Natural number Combinatorics Mathematics Representation (politics) Set (abstract data type) Real number Binary logarithm Discrete mathematics Computer science Law Programming language Politics Political science

Metadata

Type: preprint
Language: en
Landing Page: https://doi.org/10.5486/pmd.2018.8175
OA Status: green
Cited By: 9
References: 2
Related Works: 12
OpenAlex ID: https://openalex.org/W206919988

All OpenAlex metadata

Raw OpenAlex JSON

OpenAlex ID: https://openalex.org/W206919988

Canonical identifier for this work in OpenAlex
DOI: https://doi.org/10.5486/pmd.2018.8175

Digital Object Identifier
Title: On additive representation functions

Work title
Type: preprint

OpenAlex work type
Language: en

Primary language
Publication year: 2018

Year of publication
Publication date: 2018-07-01

Full publication date if available
Authors: Yong-Gao Chen, Hui Lv

List of authors in order
Landing page: https://doi.org/10.5486/pmd.2018.8175

Publisher landing page
Open access: Yes

Whether a free full text is available
OA status: green

Open access status per OpenAlex
OA URL: https://arxiv.org/pdf/1711.00186

Direct OA link when available
Concepts: Natural number, Combinatorics, Mathematics, Representation (politics), Set (abstract data type), Real number, Binary logarithm, Discrete mathematics, Computer science, Law, Programming language, Politics, Political science

Top concepts (fields/topics) attached by OpenAlex
Cited by: 9

Total citation count in OpenAlex
Citations by year (recent): 2025: 1, 2023: 1, 2021: 2, 2020: 2, 2016: 1

Per-year citation counts (last 5 years)
References (count): 2

Number of works referenced by this work
Related works (count): 12

Other works algorithmically related by OpenAlex

Full payload

id	https://openalex.org/W206919988
doi	https://doi.org/10.5486/pmd.2018.8175
ids.doi	https://doi.org/10.48550/arxiv.1711.00186
ids.mag	206919988
ids.openalex	https://openalex.org/W206919988
fwci	2.45887446
type	preprint
title	On additive representation functions
biblio.issue	1-2
biblio.volume	93
biblio.last_page	213
biblio.first_page	205
topics[0].id	https://openalex.org/T11329
topics[0].field.id	https://openalex.org/fields/26
topics[0].field.display_name	Mathematics
topics[0].score	0.9986000061035156
topics[0].domain.id	https://openalex.org/domains/3
topics[0].domain.display_name	Physical Sciences
topics[0].subfield.id	https://openalex.org/subfields/2607
topics[0].subfield.display_name	Discrete Mathematics and Combinatorics
topics[0].display_name	Limits and Structures in Graph Theory
topics[1].id	https://openalex.org/T11567
topics[1].field.id	https://openalex.org/fields/17
topics[1].field.display_name	Computer Science
topics[1].score	0.984000027179718
topics[1].domain.id	https://openalex.org/domains/3
topics[1].domain.display_name	Physical Sciences
topics[1].subfield.id	https://openalex.org/subfields/1703
topics[1].subfield.display_name	Computational Theory and Mathematics
topics[1].display_name	semigroups and automata theory
topics[2].id	https://openalex.org/T10374
topics[2].field.id	https://openalex.org/fields/17
topics[2].field.display_name	Computer Science
topics[2].score	0.9595999717712402
topics[2].domain.id	https://openalex.org/domains/3
topics[2].domain.display_name	Physical Sciences
topics[2].subfield.id	https://openalex.org/subfields/1703
topics[2].subfield.display_name	Computational Theory and Mathematics
topics[2].display_name	Advanced Graph Theory Research
is_xpac	False
apc_list
apc_paid
concepts[0].id	https://openalex.org/C184264201
concepts[0].level	2
concepts[0].score	0.637600839138031
concepts[0].wikidata	https://www.wikidata.org/wiki/Q21199
concepts[0].display_name	Natural number
concepts[1].id	https://openalex.org/C114614502
concepts[1].level	1
concepts[1].score	0.6307093501091003
concepts[1].wikidata	https://www.wikidata.org/wiki/Q76592
concepts[1].display_name	Combinatorics
concepts[2].id	https://openalex.org/C33923547
concepts[2].level	0
concepts[2].score	0.6132184267044067
concepts[2].wikidata	https://www.wikidata.org/wiki/Q395
concepts[2].display_name	Mathematics
concepts[3].id	https://openalex.org/C2776359362
concepts[3].level	3
concepts[3].score	0.47944679856300354
concepts[3].wikidata	https://www.wikidata.org/wiki/Q2145286
concepts[3].display_name	Representation (politics)
concepts[4].id	https://openalex.org/C177264268
concepts[4].level	2
concepts[4].score	0.46388158202171326
concepts[4].wikidata	https://www.wikidata.org/wiki/Q1514741
concepts[4].display_name	Set (abstract data type)
concepts[5].id	https://openalex.org/C94020503
concepts[5].level	2
concepts[5].score	0.46286165714263916
concepts[5].wikidata	https://www.wikidata.org/wiki/Q12916
concepts[5].display_name	Real number
concepts[6].id	https://openalex.org/C63553672
concepts[6].level	2
concepts[6].score	0.41660380363464355
concepts[6].wikidata	https://www.wikidata.org/wiki/Q581168
concepts[6].display_name	Binary logarithm
concepts[7].id	https://openalex.org/C118615104
concepts[7].level	1
concepts[7].score	0.3819192349910736
concepts[7].wikidata	https://www.wikidata.org/wiki/Q121416
concepts[7].display_name	Discrete mathematics
concepts[8].id	https://openalex.org/C41008148
concepts[8].level	0
concepts[8].score	0.06806972622871399
concepts[8].wikidata	https://www.wikidata.org/wiki/Q21198
concepts[8].display_name	Computer science
concepts[9].id	https://openalex.org/C199539241
concepts[9].level	1
concepts[9].score	0.0
concepts[9].wikidata	https://www.wikidata.org/wiki/Q7748
concepts[9].display_name	Law
concepts[10].id	https://openalex.org/C199360897
concepts[10].level	1
concepts[10].score	0.0
concepts[10].wikidata	https://www.wikidata.org/wiki/Q9143
concepts[10].display_name	Programming language
concepts[11].id	https://openalex.org/C94625758
concepts[11].level	2
concepts[11].score	0.0
concepts[11].wikidata	https://www.wikidata.org/wiki/Q7163
concepts[11].display_name	Politics
concepts[12].id	https://openalex.org/C17744445
concepts[12].level	0
concepts[12].score	0.0
concepts[12].wikidata	https://www.wikidata.org/wiki/Q36442
concepts[12].display_name	Political science
keywords[0].id	https://openalex.org/keywords/natural-number
keywords[0].score	0.637600839138031
keywords[0].display_name	Natural number
keywords[1].id	https://openalex.org/keywords/combinatorics
keywords[1].score	0.6307093501091003
keywords[1].display_name	Combinatorics
keywords[2].id	https://openalex.org/keywords/mathematics
keywords[2].score	0.6132184267044067
keywords[2].display_name	Mathematics
keywords[3].id	https://openalex.org/keywords/representation
keywords[3].score	0.47944679856300354
keywords[3].display_name	Representation (politics)
keywords[4].id	https://openalex.org/keywords/set
keywords[4].score	0.46388158202171326
keywords[4].display_name	Set (abstract data type)
keywords[5].id	https://openalex.org/keywords/real-number
keywords[5].score	0.46286165714263916
keywords[5].display_name	Real number
keywords[6].id	https://openalex.org/keywords/binary-logarithm
keywords[6].score	0.41660380363464355
keywords[6].display_name	Binary logarithm
keywords[7].id	https://openalex.org/keywords/discrete-mathematics
keywords[7].score	0.3819192349910736
keywords[7].display_name	Discrete mathematics
keywords[8].id	https://openalex.org/keywords/computer-science
keywords[8].score	0.06806972622871399
keywords[8].display_name	Computer science
language	en
locations[0].id	doi:10.5486/pmd.2018.8175
locations[0].is_oa	False
locations[0].source.id	https://openalex.org/S203572503
locations[0].source.issn	0033-3883, 2064-2849
locations[0].source.type	journal
locations[0].source.is_oa	False
locations[0].source.issn_l	0033-3883
locations[0].source.is_core	True
locations[0].source.is_in_doaj	False
locations[0].source.display_name	Publicationes Mathematicae Debrecen
locations[0].source.host_organization	https://openalex.org/P4310315439
locations[0].source.host_organization_name	University of Debrecen
locations[0].source.host_organization_lineage	https://openalex.org/P4310315439
locations[0].source.host_organization_lineage_names	University of Debrecen
locations[0].license
locations[0].pdf_url
locations[0].version	publishedVersion
locations[0].raw_type	journal-article
locations[0].license_id
locations[0].is_accepted	True
locations[0].is_published	True
locations[0].raw_source_name	Publicationes Mathematicae Debrecen
locations[0].landing_page_url	https://doi.org/10.5486/pmd.2018.8175
locations[1].id	pmh:oai:arXiv.org:1711.00186
locations[1].is_oa	True
locations[1].source.id	https://openalex.org/S4306400194
locations[1].source.issn
locations[1].source.type	repository
locations[1].source.is_oa	True
locations[1].source.issn_l
locations[1].source.is_core	False
locations[1].source.is_in_doaj	False
locations[1].source.display_name	arXiv (Cornell University)
locations[1].source.host_organization	https://openalex.org/I205783295
locations[1].source.host_organization_name	Cornell University
locations[1].source.host_organization_lineage	https://openalex.org/I205783295
locations[1].license
locations[1].pdf_url	https://arxiv.org/pdf/1711.00186
locations[1].version	submittedVersion
locations[1].raw_type	text
locations[1].license_id
locations[1].is_accepted	False
locations[1].is_published	False
locations[1].raw_source_name
locations[1].landing_page_url	http://arxiv.org/abs/1711.00186
locations[2].id	mag:206919988
locations[2].is_oa	True
locations[2].source.id	https://openalex.org/S4306400194
locations[2].source.issn
locations[2].source.type	repository
locations[2].source.is_oa	True
locations[2].source.issn_l
locations[2].source.is_core	False
locations[2].source.is_in_doaj	False
locations[2].source.display_name	arXiv (Cornell University)
locations[2].source.host_organization	https://openalex.org/I205783295
locations[2].source.host_organization_name	Cornell University
locations[2].source.host_organization_lineage	https://openalex.org/I205783295
locations[2].license
locations[2].pdf_url
locations[2].version	submittedVersion
locations[2].raw_type
locations[2].license_id
locations[2].is_accepted	False
locations[2].is_published	False
locations[2].raw_source_name	arXiv (Cornell University)
locations[2].landing_page_url	http://export.arxiv.org/pdf/1711.00186
locations[3].id	doi:10.48550/arxiv.1711.00186
locations[3].is_oa	True
locations[3].source.id	https://openalex.org/S4306400194
locations[3].source.issn
locations[3].source.type	repository
locations[3].source.is_oa	True
locations[3].source.issn_l
locations[3].source.is_core	False
locations[3].source.is_in_doaj	False
locations[3].source.display_name	arXiv (Cornell University)
locations[3].source.host_organization	https://openalex.org/I205783295
locations[3].source.host_organization_name	Cornell University
locations[3].source.host_organization_lineage	https://openalex.org/I205783295
locations[3].license
locations[3].pdf_url
locations[3].version
locations[3].raw_type	article-journal
locations[3].license_id
locations[3].is_accepted	False
locations[3].is_published
locations[3].raw_source_name
locations[3].landing_page_url	https://doi.org/10.48550/arxiv.1711.00186
indexed_in	arxiv, crossref, datacite
authorships[0].author.id	https://openalex.org/A5071548410
authorships[0].author.orcid	https://orcid.org/0000-0002-8775-431X
authorships[0].author.display_name	Yong-Gao Chen
authorships[0].author_position	first
authorships[0].raw_author_name	Yong-Gao Chen
authorships[0].is_corresponding	False
authorships[1].author.id	https://openalex.org/A5048910316
authorships[1].author.orcid
authorships[1].author.display_name	Hui Lv
authorships[1].author_position	last
authorships[1].raw_author_name	Hui Lv
authorships[1].is_corresponding	False
has_content.pdf	False
has_content.grobid_xml	False
is_paratext	False
open_access.is_oa	True
open_access.oa_url	https://arxiv.org/pdf/1711.00186
open_access.oa_status	green
open_access.any_repository_has_fulltext	False
created_date	2025-10-10T00:00:00
display_name	On additive representation functions
has_fulltext	False
is_retracted	False
updated_date	2025-11-06T03:46:38.306776
primary_topic.id	https://openalex.org/T11329
primary_topic.field.id	https://openalex.org/fields/26
primary_topic.field.display_name	Mathematics
primary_topic.score	0.9986000061035156
primary_topic.domain.id	https://openalex.org/domains/3
primary_topic.domain.display_name	Physical Sciences
primary_topic.subfield.id	https://openalex.org/subfields/2607
primary_topic.subfield.display_name	Discrete Mathematics and Combinatorics
primary_topic.display_name	Limits and Structures in Graph Theory
related_works	https://openalex.org/W1975599766, https://openalex.org/W2154155596, https://openalex.org/W1991321293, https://openalex.org/W2428137983, https://openalex.org/W827032062, https://openalex.org/W2023929122, https://openalex.org/W2047478385, https://openalex.org/W2591116635, https://openalex.org/W2923525384, https://openalex.org/W2790353619, https://openalex.org/W870803206, https://openalex.org/W141439497
cited_by_count	9
counts_by_year[0].year	2025
counts_by_year[0].cited_by_count	1
counts_by_year[1].year	2023
counts_by_year[1].cited_by_count	1
counts_by_year[2].year	2021
counts_by_year[2].cited_by_count	2
counts_by_year[3].year	2020
counts_by_year[3].cited_by_count	2
counts_by_year[4].year	2016
counts_by_year[4].cited_by_count	1
counts_by_year[5].year	2013
counts_by_year[5].cited_by_count	1
counts_by_year[6].year	2012
counts_by_year[6].cited_by_count	1
locations_count	4
best_oa_location.id	pmh:oai:arXiv.org:1711.00186
best_oa_location.is_oa	True
best_oa_location.source.id	https://openalex.org/S4306400194
best_oa_location.source.issn
best_oa_location.source.type	repository
best_oa_location.source.is_oa	True
best_oa_location.source.issn_l
best_oa_location.source.is_core	False
best_oa_location.source.is_in_doaj	False
best_oa_location.source.display_name	arXiv (Cornell University)
best_oa_location.source.host_organization	https://openalex.org/I205783295
best_oa_location.source.host_organization_name	Cornell University
best_oa_location.source.host_organization_lineage	https://openalex.org/I205783295
best_oa_location.license
best_oa_location.pdf_url	https://arxiv.org/pdf/1711.00186
best_oa_location.version	submittedVersion
best_oa_location.raw_type	text
best_oa_location.license_id
best_oa_location.is_accepted	False
best_oa_location.is_published	False
best_oa_location.raw_source_name
best_oa_location.landing_page_url	http://arxiv.org/abs/1711.00186
primary_location.id	doi:10.5486/pmd.2018.8175
primary_location.is_oa	False
primary_location.source.id	https://openalex.org/S203572503
primary_location.source.issn	0033-3883, 2064-2849
primary_location.source.type	journal
primary_location.source.is_oa	False
primary_location.source.issn_l	0033-3883
primary_location.source.is_core	True
primary_location.source.is_in_doaj	False
primary_location.source.display_name	Publicationes Mathematicae Debrecen
primary_location.source.host_organization	https://openalex.org/P4310315439
primary_location.source.host_organization_name	University of Debrecen
primary_location.source.host_organization_lineage	https://openalex.org/P4310315439
primary_location.source.host_organization_lineage_names	University of Debrecen
primary_location.license
primary_location.pdf_url
primary_location.version	publishedVersion
primary_location.raw_type	journal-article
primary_location.license_id
primary_location.is_accepted	True
primary_location.is_published	True
primary_location.raw_source_name	Publicationes Mathematicae Debrecen
primary_location.landing_page_url	https://doi.org/10.5486/pmd.2018.8175
publication_date	2018-07-01
publication_year	2018
referenced_works	https://openalex.org/W1980995710, https://openalex.org/W1551037730
referenced_works_count	2
abstract_inverted_index.1$	56
abstract_inverted_index.12	66
abstract_inverted_index.A,	27
abstract_inverted_index.In	47
abstract_inverted_index.an	3
abstract_inverted_index.be	2, 32
abstract_inverted_index.if	53
abstract_inverted_index.in	37
abstract_inverted_index.of	6, 18, 20, 35
abstract_inverted_index.or	43
abstract_inverted_index.to	45
abstract_inverted_index.we	50
abstract_inverted_index.$A$	1, 38
abstract_inverted_index.$a,	25
abstract_inverted_index.For	9
abstract_inverted_index.Let	0, 30
abstract_inverted_index.all	58, 71
abstract_inverted_index.are	40
abstract_inverted_index.b$.	29
abstract_inverted_index.for	57, 70
abstract_inverted_index.let	12
abstract_inverted_index.n)$	14
abstract_inverted_index.set	5
abstract_inverted_index.the	16, 21, 33
abstract_inverted_index.$n$,	62
abstract_inverted_index.$x$.	46, 74
abstract_inverted_index.a\le	28
abstract_inverted_index.b\in	26
abstract_inverted_index.less	41
abstract_inverted_index.than	42
abstract_inverted_index.then	63
abstract_inverted_index.this	48
abstract_inverted_index.with	24
abstract_inverted_index.$n\in	10
abstract_inverted_index.$r(A,	13, 54
abstract_inverted_index.(\log	67
abstract_inverted_index.\frac	65
abstract_inverted_index.equal	44
abstract_inverted_index.large	60, 73
abstract_inverted_index.prove	51
abstract_inverted_index.that,	52
abstract_inverted_index.which	39
abstract_inverted_index.x)^2$	69
abstract_inverted_index.denote	15
abstract_inverted_index.number	17, 34
abstract_inverted_index.paper,	49
abstract_inverted_index.$n=a+b$	23
abstract_inverted_index.n)\not=	55
abstract_inverted_index.natural	7
abstract_inverted_index.$\|A(x)\|$	31
abstract_inverted_index.equation	22
abstract_inverted_index.infinite	4
abstract_inverted_index.integers	36, 61
abstract_inverted_index.numbers.	8
abstract_inverted_index.solutions	19
abstract_inverted_index.x/\log\log	68
abstract_inverted_index.$\|A(x)\|>	64
abstract_inverted_index.\mathbb{N}$,	11
abstract_inverted_index.sufficiently	59, 72
cited_by_percentile_year.max	96
cited_by_percentile_year.min	89
countries_distinct_count	0
institutions_distinct_count	2
citation_normalized_percentile.value	0.85970915
citation_normalized_percentile.is_in_top_1_percent	False
citation_normalized_percentile.is_in_top_10_percent	False