Arithmetic Progressions in Sets of Fractional Dimension

• Published in: Geometric and functional analysis Vol. 19; no. 2; pp. 429 - 456
• Main Authors: Łaba, Izabella, Pramanik, Malabika
• Format: Journal Article
• Language: English
• Published: Basel SP Birkhäuser Verlag Basel 01.09.2009
• Subjects: Analysis; Mathematics; Mathematics and Statistics; Salem sets; 42A45; Arithmetic progressions; Hausdorff dimension; 42A32; 28A78; 42A38; 11B25; restriction estimates

Let 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.