A nonlinear projection theorem for Assouad dimension and applications
| Autor: | Jonathan Fraser |
|---|---|
| Přispěvatelé: | EPSRC, The Leverhulme Trust, University of St Andrews. Pure Mathematics, University of St Andrews. Centre for Interdisciplinary Research in Computational Algebra |
| Rok vydání: | 2020 |
| Předmět: |
Mathematics - Differential Geometry
Assouad dimension Distance sets General Mathematics T-NDAS Hausdorff dimension Metric Geometry (math.MG) Dynamical Systems (math.DS) Mathematics - Metric Geometry Differential Geometry (math.DG) Mathematics - Classical Analysis and ODEs 28A80 28A78 MCP Classical Analysis and ODEs (math.CA) FOS: Mathematics Radial projections Exceptional set Mathematics::Metric Geometry Sum-product theorem QA Mathematics Mathematics - Dynamical Systems QA Nonlinear projections |
| DOI: | 10.48550/arxiv.2004.12001 |
| Popis: | We prove a general nonlinear projection theorem for Assouad dimension. This theorem has several applications including to distance sets, radial projections, and sum-product phenomena. In the setting of distance sets we are able to completely resolve the planar distance set problem for Assouad dimension, both dealing with the awkward `critical case' and providing sharp estimates for sets with Assouad dimension less than 1. In the higher dimensional setting we connect the problem to the dimension of the set of exceptions in a related (orthogonal) projection theorem. We also obtain results on pinned distance sets and our results still hold when the distances are taken with respect to a sufficiently curved norm. As another application we prove a radial projection theorem for Assouad dimension with sharp estimates on the Hausdorff dimension of the exceptional set. Comment: 21 pages |
| Databáze: | OpenAIRE |
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