Small Furstenberg sets
For α in (0, 1], a subset E of R2 is called a Furstenberg set of type α or Fα-set if for each direction e in the unit circle there is a line segment ℓe in the direction of e such that the Hausdorff dimension of the set E∩ℓe is greater than or equal to α. In this paper we use generalized Hausdorff me...
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2013
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| Acceso en línea: | http://hdl.handle.net/20.500.12110/paper_0022247X_v400_n2_p475_Molter |
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paperaa:paper_0022247X_v400_n2_p475_Molter2023-06-12T16:44:14Z Small Furstenberg sets J. Math. Anal. Appl. 2013;400(2):475-486 Molter, U. Rela, E. Dimension function Furstenberg sets Hausdorff dimension Jarník's theorems For α in (0, 1], a subset E of R2 is called a Furstenberg set of type α or Fα-set if for each direction e in the unit circle there is a line segment ℓe in the direction of e such that the Hausdorff dimension of the set E∩ℓe is greater than or equal to α. In this paper we use generalized Hausdorff measures to give estimates on the size of these sets. Our main result is to obtain a sharp dimension estimate for a whole class of zero-dimensional Furstenberg type sets. Namely, for hγ(x)=log-γ(1x), γ>0, we construct a set Eγ∈Fhγ of Hausdorff dimension not greater than 12. Since in a previous work we showed that 12 is a lower bound for the Hausdorff dimension of any E∈Fhγ, with the present construction, the value 12 is sharp for the whole class of Furstenberg sets associated to the zero dimensional functionshγ. © 2012 Elsevier Ltd. Fil:Molter, U. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. Fil:Rela, E. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. 2013 info:eu-repo/semantics/article info:ar-repo/semantics/artículo info:eu-repo/semantics/publishedVersion application/pdf eng info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by/2.5/ar http://hdl.handle.net/20.500.12110/paper_0022247X_v400_n2_p475_Molter |
| institution |
Universidad de Buenos Aires |
| institution_str |
I-28 |
| repository_str |
R-134 |
| collection |
Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) |
| language |
Inglés |
| orig_language_str_mv |
eng |
| topic |
Dimension function Furstenberg sets Hausdorff dimension Jarník's theorems |
| spellingShingle |
Dimension function Furstenberg sets Hausdorff dimension Jarník's theorems Molter, U. Rela, E. Small Furstenberg sets |
| topic_facet |
Dimension function Furstenberg sets Hausdorff dimension Jarník's theorems |
| description |
For α in (0, 1], a subset E of R2 is called a Furstenberg set of type α or Fα-set if for each direction e in the unit circle there is a line segment ℓe in the direction of e such that the Hausdorff dimension of the set E∩ℓe is greater than or equal to α. In this paper we use generalized Hausdorff measures to give estimates on the size of these sets. Our main result is to obtain a sharp dimension estimate for a whole class of zero-dimensional Furstenberg type sets. Namely, for hγ(x)=log-γ(1x), γ>0, we construct a set Eγ∈Fhγ of Hausdorff dimension not greater than 12. Since in a previous work we showed that 12 is a lower bound for the Hausdorff dimension of any E∈Fhγ, with the present construction, the value 12 is sharp for the whole class of Furstenberg sets associated to the zero dimensional functionshγ. © 2012 Elsevier Ltd. |
| format |
Artículo Artículo publishedVersion |
| author |
Molter, U. Rela, E. |
| author_facet |
Molter, U. Rela, E. |
| author_sort |
Molter, U. |
| title |
Small Furstenberg sets |
| title_short |
Small Furstenberg sets |
| title_full |
Small Furstenberg sets |
| title_fullStr |
Small Furstenberg sets |
| title_full_unstemmed |
Small Furstenberg sets |
| title_sort |
small furstenberg sets |
| publishDate |
2013 |
| url |
http://hdl.handle.net/20.500.12110/paper_0022247X_v400_n2_p475_Molter |
| work_keys_str_mv |
AT molteru smallfurstenbergsets AT relae smallfurstenbergsets |
| _version_ |
1769810091303763968 |