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The Behaviour of the Spectral Counting Function for a Family of Sets with Fractal Boundaries
Author(s) -
Fawkes James
Publication year - 1997
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s002461079600467x
Subject(s) - mathematics , box counting , irrational number , disjoint sets , function (biology) , boundary (topology) , disjoint union (topology) , dimension (graph theory) , fractal , fractal dimension , open set , pure mathematics , mathematical analysis , combinatorics , fractal analysis , geometry , evolutionary biology , biology
We construct a one‐parameter family of sets in ℝ 2 generated by a disjoint union of open squares. We study the spectral counting function associated to a variational Dirichlet eigenvalue problem on a set from this family and show that the spectral asymptotics depend not only on the Minkowski dimension of the boundary, but also on whether the values of a specific function of the parameter are rational or irrational. Furthermore, we significantly sharpen the results in the rational case.