Arithmetic progressions in sets of fractional dimension

Izabella Laba, Malabika Pramanik

Published 2007-12-22, updated 2008-01-11Version 2

Let $E\subset\rr$ be a closed set of Hausdorff dimension $\alpha$. We prove that if $\alpha$ is sufficiently close to 1, and if $E$ supports a probabilistic measure obeying appropriate dimensionality and Fourier decay conditions, then $E$ contains non-trivial 3-term arithmetic progressions.