Disintegration results for fractal measures and applications to Diophantine approximation Article Swipe

PDF

Simon Baker ·

YOU? · · 2025 · Open Access · · DOI: https://doi.org/10.48550/arxiv.2501.09599

In this paper we prove disintegration results for self-conformal measures and affinely irreducible self-similar measures. The measures appearing in the disintegration resemble self-conformal/self-similar measures for iterated function systems satisfying the strong separation condition. As an application of our results, we prove the following Diophantine statements: 1. Using a result of Pollington and Velani, we show that if $μ$ is a self-conformal measure in $\mathbb{R}$ or an affinely irreducible self-similar measure, then there exists $α>0$ such that for all $β>α$ we have $$μ\left(\left\{\mathbf{x}\in \mathbb{R}^{d}:\max_{1\leq i\leq d}|x_{i}-p_i/q|\leq \frac{1}{q^{\frac{d+1}{d}}(\log q)^β}\textrm{ for i.m. }(p_1,\ldots,p_d,q)\in \mathbb{Z}^{d}\times \mathbb{N}\right\}\right)=0.$$ 2. Using a result of Kleinbock and Weiss, we show that if $μ$ is an affinely irreducible self-similar measure, then $μ$ almost every $\mathbf{x}$ is not a singular vector.

Related Topics

Fractal

Mathematics

Mathematical Analysis

Discrete Mathematics

Physics

Concepts

Fractal Diophantine approximation Diophantine equation Mathematics Diophantine set Applied mathematics Statistical physics Mathematical analysis Discrete mathematics Physics

Metadata

Type: preprint
Language: en
Landing Page: http://arxiv.org/abs/2501.09599
PDF: https://arxiv.org/pdf/2501.09599
OA Status: green
Related Works: 10
OpenAlex ID: https://openalex.org/W4406549096

All OpenAlex metadata

Raw OpenAlex JSON

OpenAlex ID: https://openalex.org/W4406549096

Canonical identifier for this work in OpenAlex
DOI: https://doi.org/10.48550/arxiv.2501.09599

Digital Object Identifier
Title: Disintegration results for fractal measures and applications to Diophantine approximation

Work title
Type: preprint

OpenAlex work type
Language: en

Primary language
Publication year: 2025

Year of publication
Publication date: 2025-01-16

Full publication date if available
Authors: Simon Baker

List of authors in order
Landing page: https://arxiv.org/abs/2501.09599

Publisher landing page
PDF URL: https://arxiv.org/pdf/2501.09599

Direct link to full text PDF
Open access: Yes

Whether a free full text is available
OA status: green

Open access status per OpenAlex
OA URL: https://arxiv.org/pdf/2501.09599

Direct OA link when available
Concepts: Fractal, Diophantine approximation, Diophantine equation, Mathematics, Diophantine set, Applied mathematics, Statistical physics, Mathematical analysis, Discrete mathematics, Physics

Top concepts (fields/topics) attached by OpenAlex
Cited by: 0

Total citation count in OpenAlex
Related works (count): 10

Other works algorithmically related by OpenAlex

Full payload

id	https://openalex.org/W4406549096
doi	https://doi.org/10.48550/arxiv.2501.09599
ids.doi	https://doi.org/10.48550/arxiv.2501.09599
ids.openalex	https://openalex.org/W4406549096
fwci
type	preprint
title	Disintegration results for fractal measures and applications to Diophantine approximation
biblio.issue
biblio.volume
biblio.last_page
biblio.first_page
topics[0].id	https://openalex.org/T10588
topics[0].field.id	https://openalex.org/fields/26
topics[0].field.display_name	Mathematics
topics[0].score	0.9988999962806702
topics[0].domain.id	https://openalex.org/domains/3
topics[0].domain.display_name	Physical Sciences
topics[0].subfield.id	https://openalex.org/subfields/2610
topics[0].subfield.display_name	Mathematical Physics
topics[0].display_name	Mathematical Dynamics and Fractals
topics[1].id	https://openalex.org/T12516
topics[1].field.id	https://openalex.org/fields/31
topics[1].field.display_name	Physics and Astronomy
topics[1].score	0.9757999777793884
topics[1].domain.id	https://openalex.org/domains/3
topics[1].domain.display_name	Physical Sciences
topics[1].subfield.id	https://openalex.org/subfields/3109
topics[1].subfield.display_name	Statistical and Nonlinear Physics
topics[1].display_name	Advanced Mathematical Theories and Applications
topics[2].id	https://openalex.org/T13234
topics[2].field.id	https://openalex.org/fields/26
topics[2].field.display_name	Mathematics
topics[2].score	0.9502999782562256
topics[2].domain.id	https://openalex.org/domains/3
topics[2].domain.display_name	Physical Sciences
topics[2].subfield.id	https://openalex.org/subfields/2610
topics[2].subfield.display_name	Mathematical Physics
topics[2].display_name	advanced mathematical theories
is_xpac	False
apc_list
apc_paid
concepts[0].id	https://openalex.org/C40636538
concepts[0].level	2
concepts[0].score	0.7805885076522827
concepts[0].wikidata	https://www.wikidata.org/wiki/Q81392
concepts[0].display_name	Fractal
concepts[1].id	https://openalex.org/C102966492
concepts[1].level	3
concepts[1].score	0.644371509552002
concepts[1].wikidata	https://www.wikidata.org/wiki/Q1227061
concepts[1].display_name	Diophantine approximation
concepts[2].id	https://openalex.org/C206530611
concepts[2].level	2
concepts[2].score	0.602832019329071
concepts[2].wikidata	https://www.wikidata.org/wiki/Q905896
concepts[2].display_name	Diophantine equation
concepts[3].id	https://openalex.org/C33923547
concepts[3].level	0
concepts[3].score	0.5242645740509033
concepts[3].wikidata	https://www.wikidata.org/wiki/Q395
concepts[3].display_name	Mathematics
concepts[4].id	https://openalex.org/C40621787
concepts[4].level	3
concepts[4].score	0.42842841148376465
concepts[4].wikidata	https://www.wikidata.org/wiki/Q16251580
concepts[4].display_name	Diophantine set
concepts[5].id	https://openalex.org/C28826006
concepts[5].level	1
concepts[5].score	0.39291882514953613
concepts[5].wikidata	https://www.wikidata.org/wiki/Q33521
concepts[5].display_name	Applied mathematics
concepts[6].id	https://openalex.org/C121864883
concepts[6].level	1
concepts[6].score	0.36717188358306885
concepts[6].wikidata	https://www.wikidata.org/wiki/Q677916
concepts[6].display_name	Statistical physics
concepts[7].id	https://openalex.org/C134306372
concepts[7].level	1
concepts[7].score	0.21674421429634094
concepts[7].wikidata	https://www.wikidata.org/wiki/Q7754
concepts[7].display_name	Mathematical analysis
concepts[8].id	https://openalex.org/C118615104
concepts[8].level	1
concepts[8].score	0.20185807347297668
concepts[8].wikidata	https://www.wikidata.org/wiki/Q121416
concepts[8].display_name	Discrete mathematics
concepts[9].id	https://openalex.org/C121332964
concepts[9].level	0
concepts[9].score	0.19064176082611084
concepts[9].wikidata	https://www.wikidata.org/wiki/Q413
concepts[9].display_name	Physics
keywords[0].id	https://openalex.org/keywords/fractal
keywords[0].score	0.7805885076522827
keywords[0].display_name	Fractal
keywords[1].id	https://openalex.org/keywords/diophantine-approximation
keywords[1].score	0.644371509552002
keywords[1].display_name	Diophantine approximation
keywords[2].id	https://openalex.org/keywords/diophantine-equation
keywords[2].score	0.602832019329071
keywords[2].display_name	Diophantine equation
keywords[3].id	https://openalex.org/keywords/mathematics
keywords[3].score	0.5242645740509033
keywords[3].display_name	Mathematics
keywords[4].id	https://openalex.org/keywords/diophantine-set
keywords[4].score	0.42842841148376465
keywords[4].display_name	Diophantine set
keywords[5].id	https://openalex.org/keywords/applied-mathematics
keywords[5].score	0.39291882514953613
keywords[5].display_name	Applied mathematics
keywords[6].id	https://openalex.org/keywords/statistical-physics
keywords[6].score	0.36717188358306885
keywords[6].display_name	Statistical physics
keywords[7].id	https://openalex.org/keywords/mathematical-analysis
keywords[7].score	0.21674421429634094
keywords[7].display_name	Mathematical analysis
keywords[8].id	https://openalex.org/keywords/discrete-mathematics
keywords[8].score	0.20185807347297668
keywords[8].display_name	Discrete mathematics
keywords[9].id	https://openalex.org/keywords/physics
keywords[9].score	0.19064176082611084
keywords[9].display_name	Physics
language	en
locations[0].id	pmh:oai:arXiv.org:2501.09599
locations[0].is_oa	True
locations[0].source.id	https://openalex.org/S4306400194
locations[0].source.issn
locations[0].source.type	repository
locations[0].source.is_oa	True
locations[0].source.issn_l
locations[0].source.is_core	False
locations[0].source.is_in_doaj	False
locations[0].source.display_name	arXiv (Cornell University)
locations[0].source.host_organization	https://openalex.org/I205783295
locations[0].source.host_organization_name	Cornell University
locations[0].source.host_organization_lineage	https://openalex.org/I205783295
locations[0].license
locations[0].pdf_url	https://arxiv.org/pdf/2501.09599
locations[0].version	submittedVersion
locations[0].raw_type	text
locations[0].license_id
locations[0].is_accepted	False
locations[0].is_published	False
locations[0].raw_source_name
locations[0].landing_page_url	http://arxiv.org/abs/2501.09599
locations[1].id	doi:10.48550/arxiv.2501.09599
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
locations[1].version
locations[1].raw_type	article
locations[1].license_id
locations[1].is_accepted	False
locations[1].is_published
locations[1].raw_source_name
locations[1].landing_page_url	https://doi.org/10.48550/arxiv.2501.09599
indexed_in	arxiv, datacite
authorships[0].author.id	https://openalex.org/A5061943510
authorships[0].author.orcid	https://orcid.org/0000-0002-0716-6236
authorships[0].author.display_name	Simon Baker
authorships[0].author_position	first
authorships[0].raw_author_name	Baker, Simon
authorships[0].is_corresponding	True
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/2501.09599
open_access.oa_status	green
open_access.any_repository_has_fulltext	False
created_date	2025-10-10T00:00:00
display_name	Disintegration results for fractal measures and applications to Diophantine approximation
has_fulltext	False
is_retracted	False
updated_date	2025-11-06T06:51:31.235846
primary_topic.id	https://openalex.org/T10588
primary_topic.field.id	https://openalex.org/fields/26
primary_topic.field.display_name	Mathematics
primary_topic.score	0.9988999962806702
primary_topic.domain.id	https://openalex.org/domains/3
primary_topic.domain.display_name	Physical Sciences
primary_topic.subfield.id	https://openalex.org/subfields/2610
primary_topic.subfield.display_name	Mathematical Physics
primary_topic.display_name	Mathematical Dynamics and Fractals
related_works	https://openalex.org/W2323172212, https://openalex.org/W4249600134, https://openalex.org/W2757989085, https://openalex.org/W2000047385, https://openalex.org/W4391569061, https://openalex.org/W2375671194, https://openalex.org/W4390811435, https://openalex.org/W1192275632, https://openalex.org/W2391619745, https://openalex.org/W2370478068
cited_by_count	0
locations_count	2
best_oa_location.id	pmh:oai:arXiv.org:2501.09599
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/2501.09599
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/2501.09599
primary_location.id	pmh:oai:arXiv.org:2501.09599
primary_location.is_oa	True
primary_location.source.id	https://openalex.org/S4306400194
primary_location.source.issn
primary_location.source.type	repository
primary_location.source.is_oa	True
primary_location.source.issn_l
primary_location.source.is_core	False
primary_location.source.is_in_doaj	False
primary_location.source.display_name	arXiv (Cornell University)
primary_location.source.host_organization	https://openalex.org/I205783295
primary_location.source.host_organization_name	Cornell University
primary_location.source.host_organization_lineage	https://openalex.org/I205783295
primary_location.license
primary_location.pdf_url	https://arxiv.org/pdf/2501.09599
primary_location.version	submittedVersion
primary_location.raw_type	text
primary_location.license_id
primary_location.is_accepted	False
primary_location.is_published	False
primary_location.raw_source_name
primary_location.landing_page_url	http://arxiv.org/abs/2501.09599
publication_date	2025-01-16
publication_year	2025
referenced_works_count	0
abstract_inverted_index.a	47, 59, 94, 118
abstract_inverted_index.1.	45
abstract_inverted_index.2.	92
abstract_inverted_index.As	33
abstract_inverted_index.In	0
abstract_inverted_index.an	34, 65, 106
abstract_inverted_index.if	56, 103
abstract_inverted_index.in	18, 62
abstract_inverted_index.is	58, 105, 116
abstract_inverted_index.of	36, 49, 96
abstract_inverted_index.or	64
abstract_inverted_index.we	3, 39, 53, 79, 100
abstract_inverted_index.The	15
abstract_inverted_index.all	77
abstract_inverted_index.and	10, 51, 98
abstract_inverted_index.for	7, 24, 76, 87
abstract_inverted_index.not	117
abstract_inverted_index.our	37
abstract_inverted_index.the	19, 29, 41
abstract_inverted_index.$μ$	57, 104, 112
abstract_inverted_index.have	80
abstract_inverted_index.i.m.	88
abstract_inverted_index.show	54, 101
abstract_inverted_index.such	74
abstract_inverted_index.that	55, 75, 102
abstract_inverted_index.then	70, 111
abstract_inverted_index.this	1
abstract_inverted_index.Using	46, 93
abstract_inverted_index.every	114
abstract_inverted_index.i\leq	83
abstract_inverted_index.paper	2
abstract_inverted_index.prove	4, 40
abstract_inverted_index.there	71
abstract_inverted_index.Weiss,	99
abstract_inverted_index.almost	113
abstract_inverted_index.exists	72
abstract_inverted_index.result	48, 95
abstract_inverted_index.strong	30
abstract_inverted_index.Velani,	52
abstract_inverted_index.measure	61
abstract_inverted_index.results	6
abstract_inverted_index.systems	27
abstract_inverted_index.vector.	120
abstract_inverted_index.affinely	11, 66, 107
abstract_inverted_index.function	26
abstract_inverted_index.iterated	25
abstract_inverted_index.measure,	69, 110
abstract_inverted_index.measures	9, 16, 23
abstract_inverted_index.resemble	21
abstract_inverted_index.results,	38
abstract_inverted_index.singular	119
abstract_inverted_index.$α>0$	73
abstract_inverted_index.Kleinbock	97
abstract_inverted_index.appearing	17
abstract_inverted_index.following	42
abstract_inverted_index.measures.	14
abstract_inverted_index.$β>α$	78
abstract_inverted_index.Pollington	50
abstract_inverted_index.condition.	32
abstract_inverted_index.satisfying	28
abstract_inverted_index.separation	31
abstract_inverted_index.Diophantine	43
abstract_inverted_index.application	35
abstract_inverted_index.irreducible	12, 67, 108
abstract_inverted_index.statements:	44
abstract_inverted_index.$\mathbb{R}$	63
abstract_inverted_index.$\mathbf{x}$	115
abstract_inverted_index.self-similar	13, 68, 109
abstract_inverted_index.disintegration	5, 20
abstract_inverted_index.q)^β}\textrm{	86
abstract_inverted_index.self-conformal	8, 60
abstract_inverted_index.d}\|x_{i}-p_i/q\|\leq	84
abstract_inverted_index.\mathbb{Z}^{d}\times	90
abstract_inverted_index.}(p_1,\ldots,p_d,q)\in	89
abstract_inverted_index.\mathbb{R}^{d}:\max_{1\leq	82
abstract_inverted_index.self-conformal/self-similar	22
abstract_inverted_index.$$μ\left(\left\{\mathbf{x}\in	81
abstract_inverted_index.\mathbb{N}\right\}\right)=0.$$	91
abstract_inverted_index.\frac{1}{q^{\frac{d+1}{d}}(\log	85
cited_by_percentile_year
corresponding_author_ids	https://openalex.org/A5061943510
countries_distinct_count	0
institutions_distinct_count	1
citation_normalized_percentile