Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
  1. Home
  2. Search

Refine search

Refine search

Access:
Content type:
Publication date:
Apply   From and To filters
Subject: Show more
Journals Show more
Publishers: Show more
Societies Show more
Series: Show more
Collections: Show more

Actions for selected content:

Select all | Deselect all
  • View selected items
  • Export citations
  • Download PDF (zip)
  • Save to Kindle
  • Save to Dropbox
  • Save to Google Drive

Save Search

You can save your searches here and later view and run them again in "My saved searches".

Please provide a title, maximum of 40 characters.
Cancel
×

1 results

  • 14 - Applications in Number Theory

  • from Part Three - Applications
  • Jonathan M. Fraser, University of St Andrews, Scotland
  • Book:
    Assouad Dimension and Fractal Geometry
    Published online:
    13 October 2020
    Print publication:
    29 October 2020, pp 215-225

  • Summary

    Given the origins and motivation for studying the Assouad dimension, the applications in embedding theory are perhaps expected. In this chapter we discuss a collection of, perhaps more surprising, applications to several distinct problems in number theory. In Section 14.1 we consider the problem offinding arithmetic progressions inside sets of integers. Recall, for example, the Erdos conjecture on arithmetic progressions, of which the celebrated Green–Tao Theorem is a special case. It turns out that the Assouad dimension can be used to prove a weak `asymptotic' version of this conjecture. In Section 14.2 we discuss some problems in Diophantine approximation, the general area concerned with how well real numbers can be approximated by rationals. Here there is a well-developed connection between estimating the Hausdorff dimension of sets of badly approximable numbers and the lower dimension. Finally, in Section 14.3 we discuss an elegant use of the Assouad dimension in problems of definability of the integers due to Hieronymi and Miller.