Packing dimension of mean porous measures
We prove that the packing dimension of any mean porous Radon measure on $\mathbb R^d$ may be estimated from above by a function which depends on mean porosity. The upper bound tends to $d-1$ as mean porosity tends to its maximum value. This result was stated in \cite{BS}, and in a weaker form in \ci...
| Main Authors: | , , , , , , |
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| Format: | Journal article |
| Language: | English |
| Published: |
2007
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| _version_ | 1797085626777993216 |
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| author | Beliaev, D Järvenpää, E Järvenpää, M Käenmäki, A Rajala, T Smirnov, S Suomala, V |
| author_facet | Beliaev, D Järvenpää, E Järvenpää, M Käenmäki, A Rajala, T Smirnov, S Suomala, V |
| author_sort | Beliaev, D |
| collection | OXFORD |
| description | We prove that the packing dimension of any mean porous Radon measure on $\mathbb R^d$ may be estimated from above by a function which depends on mean porosity. The upper bound tends to $d-1$ as mean porosity tends to its maximum value. This result was stated in \cite{BS}, and in a weaker form in \cite{JJ1}, but the proofs are not correct. Quite surprisingly, it turns out that mean porous measures are not necessarily approximable by mean porous sets. We verify this by constructing an example of a mean porous measure $\mu$ on $\mathbb R$ such that $\mu(A)=0$ for all mean porous sets $A\subset\mathbb R$. |
| first_indexed | 2024-03-07T02:11:09Z |
| format | Journal article |
| id | oxford-uuid:a0b14aee-2ade-498d-a7ac-3643616c2cdd |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T02:11:09Z |
| publishDate | 2007 |
| record_format | dspace |
| spelling | oxford-uuid:a0b14aee-2ade-498d-a7ac-3643616c2cdd2022-03-27T02:07:19ZPacking dimension of mean porous measuresJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:a0b14aee-2ade-498d-a7ac-3643616c2cddEnglishSymplectic Elements at Oxford2007Beliaev, DJärvenpää, EJärvenpää, MKäenmäki, ARajala, TSmirnov, SSuomala, VWe prove that the packing dimension of any mean porous Radon measure on $\mathbb R^d$ may be estimated from above by a function which depends on mean porosity. The upper bound tends to $d-1$ as mean porosity tends to its maximum value. This result was stated in \cite{BS}, and in a weaker form in \cite{JJ1}, but the proofs are not correct. Quite surprisingly, it turns out that mean porous measures are not necessarily approximable by mean porous sets. We verify this by constructing an example of a mean porous measure $\mu$ on $\mathbb R$ such that $\mu(A)=0$ for all mean porous sets $A\subset\mathbb R$. |
| spellingShingle | Beliaev, D Järvenpää, E Järvenpää, M Käenmäki, A Rajala, T Smirnov, S Suomala, V Packing dimension of mean porous measures |
| title | Packing dimension of mean porous measures |
| title_full | Packing dimension of mean porous measures |
| title_fullStr | Packing dimension of mean porous measures |
| title_full_unstemmed | Packing dimension of mean porous measures |
| title_short | Packing dimension of mean porous measures |
| title_sort | packing dimension of mean porous measures |
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