Geometric and Functional Analysis

, Volume 19, Issue 2, pp 429-456

First online:

Arithmetic Progressions in Sets of Fractional Dimension

  • Izabella ŁabaAffiliated withDepartment of Mathematics, University of British Columbia Email author 
  • , Malabika PramanikAffiliated withDepartment of Mathematics, University of British Columbia

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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