, Volume 19, Issue 2, pp 429-456
Date: 11 Jul 2009

Arithmetic Progressions in Sets of Fractional Dimension

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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.