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Abstract
We consider the Assouad dimensions of orthogonal projections of planar sets onto lines. Our investigation covers both general and selfsimilar sets.
For general sets, the main result is the following: if a set in the plane has Assouad dimension s ∈ [0, 2], then the projections have Assouad dimension at least min{1, s} almost surely. Compared to the famous analogue for Hausdorff dimension – namely Marstrand’s Projection Theorem – a striking difference is that the words ‘at least’cannot be dispensed with: in fact, for many planar selfsimilar sets of dimension s < 1, we prove that the Assouad dimension of projections can attain both values sand 1 for a set of directions of positive measure.
For selfsimilar sets, our investigation splits naturally into two cases: when the group of rotations is discrete, and when it is dense. In the ‘discrete rotations’ case we prove the following dichotomy for any given projection: either the Hausdorff measure is positive in the Hausdorff dimension, in which case the Hausdorff and Assouad dimensions coincide; or the Hausdorff measure is zero in the Hausdorff dimension,in which case the Assouad dimension is equal to 1. In the ‘dense rotations’ case we prove that every projection has Assouad dimension equal to one, assuming that the planar set is not a singleton.
As another application of our results, we show that there is no Falconer’s Theorem for Assouad dimension. More precisely, the Assouad dimension of a selfsimilar (or selfaffine) set is not in general almost surely constant when one randomises the translation vectors.
For general sets, the main result is the following: if a set in the plane has Assouad dimension s ∈ [0, 2], then the projections have Assouad dimension at least min{1, s} almost surely. Compared to the famous analogue for Hausdorff dimension – namely Marstrand’s Projection Theorem – a striking difference is that the words ‘at least’cannot be dispensed with: in fact, for many planar selfsimilar sets of dimension s < 1, we prove that the Assouad dimension of projections can attain both values sand 1 for a set of directions of positive measure.
For selfsimilar sets, our investigation splits naturally into two cases: when the group of rotations is discrete, and when it is dense. In the ‘discrete rotations’ case we prove the following dichotomy for any given projection: either the Hausdorff measure is positive in the Hausdorff dimension, in which case the Hausdorff and Assouad dimensions coincide; or the Hausdorff measure is zero in the Hausdorff dimension,in which case the Assouad dimension is equal to 1. In the ‘dense rotations’ case we prove that every projection has Assouad dimension equal to one, assuming that the planar set is not a singleton.
As another application of our results, we show that there is no Falconer’s Theorem for Assouad dimension. More precisely, the Assouad dimension of a selfsimilar (or selfaffine) set is not in general almost surely constant when one randomises the translation vectors.
Original language  English 

Pages (fromto)  374398 
Number of pages  25 
Journal  Proceedings of the London Mathematical Society 
Volume  114 
Issue number  2 
Early online date  17 Jan 2017 
DOIs  
Publication status  Published  Feb 2017 
Keywords
 Assouad dimension
 Projection
 Selfsimilar set
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Dive into the research topics of 'The Assouad dimensions of projections of planar sets'. Together they form a unique fingerprint.Projects
 1 Finished

Fractal Geometry and Dimension: Fractal Geometry and dimension theory
1/09/16 → 30/06/18
Project: Fellowship
Profiles

Jonathan Fraser
 School of Mathematics and Statistics  Director of Research
 Pure Mathematics  Professor
Person: Academic