Abstract
It is known that, unlike the Hausdorff dimension, the Assouad dimension of a selfsimilar set can exceed the similarity dimension if there are overlaps in the construction. Our main result is the following precise dichotomy for selfsimilar sets in the line: either the weak separation property is satisfied, in which case the Hausdorff and Assouad dimensions coincide; or the weak separation property is not satisfied, in which case the Assouad dimension is maximal (equal to one). In the first case we prove that the selfsimilar set is Ahlfors regular, and in the second case we use the fact that if the weak separation property is not satisfied, one can approximate the identity arbitrarily well in the group generated by the similarity mappings, and this allows us to build a weak tangent that contains an interval. We also obtain results in higher dimensions and provide illustrative examples showing that the ‘equality/maximal’ dichotomy does not extend to this setting.
Original language  English 

Pages (fromto)  188214 
Number of pages  27 
Journal  Advances in Mathematics 
Volume  273 
Early online date  14 Jan 2015 
DOIs  
Publication status  Published  19 Mar 2015 
Keywords
 Assouad dimension
 Selfsimilar set
 Weak separation property
 Overlaps
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Jonathan Fraser
 School of Mathematics and Statistics  Director of Research
 Pure Mathematics  Professor
 Centre for Interdisciplinary Research in Computational Algebra
Person: Academic