[Paper Review] Long progressions in sets of fractional dimension

Marc Carnovale|arXiv (Cornell University)|Aug 13, 2013

Limits and Structures in Graph Theory11 references4 citations

TL;DR

This paper establishes sufficient conditions for the existence of long arithmetic progressions in subsets of the real line with fractional Hausdorff dimension, using a higher-order Fourier dimension based on Gowers uniformity norms. It proves that if a measure on the torus has sufficiently high k-th order Fourier dimension, its support contains non-trivial k+1-term arithmetic progressions, with the set of common differences having positive Lebesgue measure.

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 $\mathbb{R}$ under consideration.

Motivation & Objective

• To determine sufficient conditions under which sparse subsets of R with fractional dimension contain long arithmetic progressions.
• To extend the Fourier restriction approach for 3-term progressions to k+1-term configurations.
• To quantify the size and regularity of the set of common differences in such progressions.
• To connect the problem to the Falconer Distance Conjecture by generalizing it to k+1-term configurations.
• To develop a framework using higher-order Fourier dimension and Gowers uniformity norms for detecting finite point configurations in fractal sets.

Proposed method

• Introduces a k+1-dimensional measure Δ^kμ derived from a singular measure μ on the torus, generalizing Gowers' uniformity norms.
• Defines the k-th order Fourier dimension via decay rates of the Fourier transform of Δ^kμ at scale-dependent frequencies.
• Applies a variant of the Gowers uniformity norm to singular measures, linking decay of the transform to the presence of arithmetic progressions.
• Uses hypergraph regularity and removal lemmas (e.g., Gowers’ hypergraph removal) to detect affine images of point configurations in dense sets.
• Applies a discretization argument to transfer results from finite abelian groups to the continuous setting via approximation of measures.
• Relies on the structure of the k-fold Cartesian product and the multilinear form associated with Δ^kμ to control configurations.

Experimental results

Research questions

• RQ1 Under what conditions on the higher-order Fourier dimension of a measure does its support contain k+1-term arithmetic progressions?
• RQ2 How can the Falconer Distance Conjecture be generalized to k+1-term configurations in fractal sets?
• RQ3 What is the size and regularity of the set of scaling parameters r for which affine images of a k+1-point configuration lie in the support of a measure?
• RQ4 Can the methods used for 3-term progressions be extended to longer arithmetic progressions using higher-order uniformity norms?
• RQ5 What is the role of Gowers uniformity norms in detecting finite point configurations in sets of fractional dimension?

Key findings

• If a measure μ on the torus has sufficiently high k-th order Fourier dimension (close to 1), then its support contains non-trivial k+1-term arithmetic progressions.
• The set of common differences (scaling parameters r) for which such progressions exist has positive Lebesgue measure, generalizing a trivial case of the Falconer Distance Conjecture.
• The paper establishes a quantitative lower bound on the number of such progressions via a discretization argument and hypergraph regularity.
• The main result extends the continuous Roth-type theorem for 3-term progressions to longer configurations using higher-order Fourier analysis.
• The framework allows for the detection of any finite point configuration (e.g., arithmetic progressions) in sets of fractional dimension under a uniformity norm decay condition.
• The proof relies on a novel application of the Gowers hypergraph removal lemma to continuous settings, transferring combinatorial results to the analysis of singular measures.