Geometric and Functional Analysis

, Volume 19, Issue 2, pp 429–456 | Cite as

Arithmetic Progressions in Sets of Fractional Dimension

Article

Abstract

Let \({E \subset\mathbb{R}}\) be a closed set of Hausdorff dimension α. Weprove that if α is sufficiently close to 1, and if E supports a probability measure obeying appropriate dimensionality and Fourier decay conditions, then E contains non-trivial 3-term arithmetic progressions.

Keywords and phrases

Arithmetic progressions Salem sets Hausdorff dimension restriction estimates 

2000 Mathematics Subject Classification

28A78 42A32 42A38 42A45 11B25 

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Copyright information

© Birkhäuser Verlag Basel/Switzerland 2009

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of British ColumbiaVancouverCanada

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