Simon’s OPUC Hausdorff dimension conjecture
We show that the Szegő matrices, associated with Verblunsky coefficients { α n } n ∈ Z + obeying ∑ n = 0 ∞ n γ | α n | 2 < ∞ for some γ ∈ ( 0 , 1 ) , are bounded for values z ∈ ∂ D outside a set of Hausdorff dimension no more than 1 - γ . In particular, the singular part of the associated probabi...
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| Veröffentlicht in: | Mathematische annalen 2022-10, Vol.384 (1-2), p.1-37 |
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| container_title | Mathematische annalen |
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| creator | Damanik, David Guo, Shuzheng Ong, Darren C. |
| description | We show that the Szegő matrices, associated with Verblunsky coefficients
{
α
n
}
n
∈
Z
+
obeying
∑
n
=
0
∞
n
γ
|
α
n
|
2
<
∞
for some
γ
∈
(
0
,
1
)
, are bounded for values
z
∈
∂
D
outside a set of Hausdorff dimension no more than
1
-
γ
. In particular, the singular part of the associated probability measure on the unit circle is supported by a set of Hausdorff dimension no more than
1
-
γ
. This proves the OPUC Hausdorff dimension conjecture of Barry Simon from 2005. |
| doi_str_mv | 10.1007/s00208-021-02283-7 |
| format | Article |
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{
α
n
}
n
∈
Z
+
obeying
∑
n
=
0
∞
n
γ
|
α
n
|
2
<
∞
for some
γ
∈
(
0
,
1
)
, are bounded for values
z
∈
∂
D
outside a set of Hausdorff dimension no more than
1
-
γ
. In particular, the singular part of the associated probability measure on the unit circle is supported by a set of Hausdorff dimension no more than
1
-
γ
. This proves the OPUC Hausdorff dimension conjecture of Barry Simon from 2005.</description><identifier>ISSN: 0025-5831</identifier><identifier>EISSN: 1432-1807</identifier><identifier>DOI: 10.1007/s00208-021-02283-7</identifier><language>eng</language><publisher>Berlin/Heidelberg: Springer Berlin Heidelberg</publisher><subject>Mathematics ; Mathematics and Statistics</subject><ispartof>Mathematische annalen, 2022-10, Vol.384 (1-2), p.1-37</ispartof><rights>The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2021</rights><rights>The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2021.</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c319t-56512cba5cac7add6c8a81c30872d70c0a0bb328054507e08b682cc41dc1b46e3</citedby><cites>FETCH-LOGICAL-c319t-56512cba5cac7add6c8a81c30872d70c0a0bb328054507e08b682cc41dc1b46e3</cites><orcidid>0000-0003-4942-4376 ; 0000-0002-3816-3652</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s00208-021-02283-7$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s00208-021-02283-7$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,776,780,27901,27902,41464,42533,51294</link.rule.ids></links><search><creatorcontrib>Damanik, David</creatorcontrib><creatorcontrib>Guo, Shuzheng</creatorcontrib><creatorcontrib>Ong, Darren C.</creatorcontrib><title>Simon’s OPUC Hausdorff dimension conjecture</title><title>Mathematische annalen</title><addtitle>Math. Ann</addtitle><description>We show that the Szegő matrices, associated with Verblunsky coefficients
{
α
n
}
n
∈
Z
+
obeying
∑
n
=
0
∞
n
γ
|
α
n
|
2
<
∞
for some
γ
∈
(
0
,
1
)
, are bounded for values
z
∈
∂
D
outside a set of Hausdorff dimension no more than
1
-
γ
. In particular, the singular part of the associated probability measure on the unit circle is supported by a set of Hausdorff dimension no more than
1
-
γ
. This proves the OPUC Hausdorff dimension conjecture of Barry Simon from 2005.</description><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><issn>0025-5831</issn><issn>1432-1807</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2022</creationdate><recordtype>article</recordtype><recordid>eNp9kMFKxDAQhoMoWFdfwFPBc3SSNE16lKKusLCC7jmk01S6uM2atAdvvoav55MYreDNwzCH-f5_4CPknMElA1BXEYCDpsBZGq4FVQckY4XglGlQhyRLd0mlFuyYnMS4BQABIDNCH_udHz7fP2K-ftjU-dJOsfWh6_K237kh9n7I0Q9bh-MU3Ck56uxLdGe_e0E2tzdP9ZKu1nf39fWKomDVSGUpGcfGSrSobNuWqK1mKEAr3ipAsNA0gmuQhQTlQDel5ogFa5E1RenEglzMvfvgXycXR7P1UxjSS8MVVEJUJehE8ZnC4GMMrjP70O9seDMMzLcWM2sxSYv50WJUCok5FBM8PLvwV_1P6gsYsmRq</recordid><startdate>20221001</startdate><enddate>20221001</enddate><creator>Damanik, David</creator><creator>Guo, Shuzheng</creator><creator>Ong, Darren C.</creator><general>Springer Berlin Heidelberg</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope><orcidid>https://orcid.org/0000-0003-4942-4376</orcidid><orcidid>https://orcid.org/0000-0002-3816-3652</orcidid></search><sort><creationdate>20221001</creationdate><title>Simon’s OPUC Hausdorff dimension conjecture</title><author>Damanik, David ; Guo, Shuzheng ; Ong, Darren C.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c319t-56512cba5cac7add6c8a81c30872d70c0a0bb328054507e08b682cc41dc1b46e3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2022</creationdate><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Damanik, David</creatorcontrib><creatorcontrib>Guo, Shuzheng</creatorcontrib><creatorcontrib>Ong, Darren C.</creatorcontrib><collection>CrossRef</collection><jtitle>Mathematische annalen</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Damanik, David</au><au>Guo, Shuzheng</au><au>Ong, Darren C.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Simon’s OPUC Hausdorff dimension conjecture</atitle><jtitle>Mathematische annalen</jtitle><stitle>Math. Ann</stitle><date>2022-10-01</date><risdate>2022</risdate><volume>384</volume><issue>1-2</issue><spage>1</spage><epage>37</epage><pages>1-37</pages><issn>0025-5831</issn><eissn>1432-1807</eissn><abstract>We show that the Szegő matrices, associated with Verblunsky coefficients
{
α
n
}
n
∈
Z
+
obeying
∑
n
=
0
∞
n
γ
|
α
n
|
2
<
∞
for some
γ
∈
(
0
,
1
)
, are bounded for values
z
∈
∂
D
outside a set of Hausdorff dimension no more than
1
-
γ
. In particular, the singular part of the associated probability measure on the unit circle is supported by a set of Hausdorff dimension no more than
1
-
γ
. This proves the OPUC Hausdorff dimension conjecture of Barry Simon from 2005.</abstract><cop>Berlin/Heidelberg</cop><pub>Springer Berlin Heidelberg</pub><doi>10.1007/s00208-021-02283-7</doi><tpages>37</tpages><orcidid>https://orcid.org/0000-0003-4942-4376</orcidid><orcidid>https://orcid.org/0000-0002-3816-3652</orcidid></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0025-5831 |
| ispartof | Mathematische annalen, 2022-10, Vol.384 (1-2), p.1-37 |
| issn | 0025-5831 1432-1807 |
| language | eng |
| recordid | cdi_proquest_journals_2709339608 |
| source | SpringerLink Journals |
| subjects | Mathematics Mathematics and Statistics |
| title | Simon’s OPUC Hausdorff dimension conjecture |
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