Arithmetic progressions in sets of fractional dimension

Abstract

Let E ⊂ 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. © 2009 Birkhäuser Verlag Basel/Switzerland.