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.
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Łaba, I., Pramanik, M. Arithmetic Progressions in Sets of Fractional Dimension. Geom. Funct. Anal. 19, 429–456 (2009). https://doi.org/10.1007/s00039-009-0003-9
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DOI: https://doi.org/10.1007/s00039-009-0003-9