Long progressions in sets of fractional dimension

arXiv1308.2919MaRDI QIDQ6244089

Marc Carnovale

Publication date: 13 August 2013

Abstract: We demonstrate $k+1$-term arithmetic progressions in certain subsets of the real line whose "higher-order Fourier dimension" is sufficiently close to 1. This Fourier dimension, introduced in previous work, is a higher-order (in the sense of Additive Combinatorics and uniformity norms) extension of the Fourier dimension of Geometric Measure Theory, and can be understood as asking that the uniformity norm of a measure, restricted to a given scale, decay as the scale increases. We further obtain quantitative information about the size and $L^p$ regularity of the set of common distances of the artihmetic progressions contained in the subsets of $mathbbR$ under consideration.

Mathematics Subject Classification ID

Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type (42A38) Length, area, volume, other geometric measure theory (28A75) Multipliers in one variable harmonic analysis (42A45) Arithmetic progressions (11B25) Hausdorff and packing measures (28A78) Trigonometric series of special types (positive coefficients, monotonic coefficients, etc.) (42A32)

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