46 research outputs found
New Hausdorff type dimensions and optimal bounds for bilipschitz invariant dimensions
We introduce a new family of fractal dimensions by restricting the set of
diameters in the coverings in the usual definition of the Hausdorff dimension.
Among others, we prove that this family contains continuum many distinct
dimensions, and they share most of the properties of the Hausdorff dimension,
which answers negatively a question of Fraser. On the other hand, we also prove
that among these new dimensions only the Hausdorff dimension behaves nicely
with respect to H\"older functions, which supports a conjecture posed by Banaji
obtained as a natural modification of Fraser's question.
We also consider the supremum of these new dimensions, which turns out to be
an other interesting notion of fractal dimension.
We prove that among those bilipschitz invariant, monotone dimensions on the
compact subsets of that agree with the similarity dimension for
the simplest self-similar sets, the modified lower dimension is the smallest
and when the Assouad dimension is the greatest, and this latter statement
is false for . This answers a question of Rutar.Comment: 22 pages, minor modifications, Question 1.5 has been remove
Valós függvénytani kutatások = Topics in Real Analysis
A tágabb Ă©rtelemben vett valĂłs fĂĽggvĂ©nytan több terĂĽletĂ©n folytattunk sikeres kutatást. JelentĹ‘s eredmĂ©nyeket Ă©rtĂĽnk számos leĂrĂł halmazelmĂ©leti, geometriai mĂ©rtĂ©kelmĂ©leti kĂ©rdĂ©sben, valamint az operátorfĂ©lcsoportok elmĂ©letĂ©ben. Kutatásaink egy rĂ©szĂ©be diákokat is bevontunk. A pályázat támogatásával nĂ©gy Ă©v alatt összesen 27 megjelent vagy elfogadott, továbbá 5 mĂ©g elbĂrálás alatt állĂł cikk szĂĽletett. EredmĂ©nyeinket számos nemzetközi konferencián is bemutattuk, többször meghĂvott elĹ‘adĂłkĂ©nt. Keleti Tamásnak Ă©s Elekes Mártonnak sikerĂĽlt D. Mauldin egy 15 Ă©ves kĂ©rdĂ©sĂ©re válaszolva megmutatni, hogy a Liouville számok halmazát ''nem lehet megmĂ©rni'', azaz a mĂ©rtĂ©ke minden eltolás invariáns Borel mĂ©rtĂ©kre nĂ©zve nulla vagy nem szigma-vĂ©ges. További geometriai mĂ©rtĂ©kelmĂ©leti eredmĂ©nyein kĂvĂĽl Keleti számos eredmĂ©nyt Ă©rt el fĂĽggvĂ©nyeknek periodikus fĂĽggvĂ©nyek összegĂ©re valĂł felbonthatĂłságával kapcsolatban. Elekes munkáiban nagy hangsĂşlyt kapott a halmazelmĂ©let, több meglepĹ‘ fĂĽggetlensĂ©gi eredmĂ©nyt igazolt. Mátrai Tamás kidolgozta a topologikus Hurewicz teszthalmazok elmĂ©letĂ©t, melynek számos alkalmazását találta. Az operátorfĂ©lcsoportok normafolytonosságával kapcsolatos kutatásaiban egy Ăşj Banach-tĂ©r konstrukciĂł segĂtsĂ©gĂ©vel ellenpĂ©ldát adott A. Pazy egy 40 Ă©ves sejtĂ©sĂ©re. Csörnyei Marianna szerzĹ‘társaival geometriai mĂ©rtĂ©kelmĂ©leti jellemzĂ©st talált Lagrange fĂĽggvĂ©nyekkel megadott variáciĂłszámĂtási problĂ©mák Ăşgynevezett univerzális szinguláris halmazaira. | Our research group carried out research work in several areas of real analysis. We obtained notable results concerning many problems in descriptive set theory, geometric measure theory and in the theory of operator semigroups. Our research was partially done in collaboration with our students. With the support of the present fellowship, within four years, we published 32 papers, 27 of which had already appeared or had been accepted for publication. We presented our results on numerous international conferences, on several occasions as invited speakers. Keleti and Elekes answered a 15-year-old problem of Mauldin by showing that the set of Liouville numbers cannot be measured: its measure with respect to any translation invariant Borel measure is either zero or non-sigma-finite. Apart from solving other geometric measure theoretic problems Keleti obtained several results about the decomposition of functions to the sum of periodic functions. The work of Elekes had more set theoretic flavor, he obtained several surprising independence results. Mátrai developed the theory of topological Hurewicz test pairs and found several applications of his theory. In his research on operator semigroups he refuted a 40-year-old conjecture of A. Pazy by inventing and using a new way to construct Banach spaces. Csörnyei, with his collaborators, found a geometric measure theoretic characterization of the universal singular sets of Lagrangians of variational problems