[Paper Review] A Non-Linear Roth Theorem for Sets of Positive Density

Ben Krause|arXiv (Cornell University)|Jan 5, 2019

Limits and Structures in Graph Theory17 references4 citations

TL;DR

This paper establishes a non-linear Roth-type theorem for sets of positive upper density in the real line, proving that for any polynomial $ P(t) $ without constant or linear terms (or more generally, non-flat curves), there exist syndetic configurations of the form $ \{x - t, x - P(t)\} $ inside such sets. Using a refined Fourier-analytic method inspired by Bourgain’s approach, the authors show that for large scales, such configurations appear with density bounded below by a constant multiple of $ \delta^2 $, where $ \delta $ is the upper density of the set.

ABSTRACT

Suppose that $A \subset \mathbb{R}$ has positive upper density, \[ \limsup_{|I| o \infty} \frac{|A \cap I|}{|I|} = δ> 0,\] and $P(t) \in \mathbb{R}[t]$ is a polynomial with no constant or linear term, or more generally a non-flat curve: a locally differentiable curve which doesn't "resemble a line" near $0$ or $\infty$. Then for any $R_0 \leq R$ sufficiently large, there exists some $x_R \in A$ so that \[ \inf_{R_0 \leq T \leq R} \frac{|\{ 0 \leq t < T : x_R - t \in A, \ x_R - P(t) \in A \}|}{T} \geq c_P \cdot δ^2 \] for some absolute constant $c_P > 0$, that depends only on $P$.

Motivation & Objective

• To extend Bourgain's Fourier-analytic method for linear Roth-type configurations to non-linear patterns involving polynomials or non-flat curves.
• To establish the existence of syndetic configurations $ \{x - t, x - P(t)\} $ in sets of positive upper density, even when $ P(t) $ has no linear or constant term.
• To generalize previous results on polynomial configurations in dense sets by incorporating non-flat curves and improving quantitative bounds.
• To provide a quantitative lower bound on the density of such configurations, dependent only on the polynomial's coefficient norm and the set's density $ \delta $.

Proposed method

• Uses a decomposition of the bilinear operator $ B_r(f,g)(x) = \frac{1}{r}\int_0^r f(x-t)g(x-P(t))\,dt $ into frequency-localized components via dyadic frequency projections.
• Applies paraproduct estimates and maximal function bounds to control error terms, particularly for high-frequency components.
• Employs a contradiction argument based on frequency localization and $ L^2 $-norm estimates on the Fourier transform of the characteristic function of the set.
• Introduces a partition of the unit interval into dyadic intervals and identifies a subset $ \mathcal{Q} $ where the bilinear operator's infimum is bounded below by $ c_P \delta^3 $.
• Applies the uncertainty principle and $ \ell^2 $-restriction estimates to show that if the bilinear form is small, then the Fourier support must be large in a controlled way.
• Uses the key estimate $ \|B_k(f_{k+m}, g_{2k+m+p})\|_{L^1} \lesssim 2^{-\epsilon m}\|f\|_2\|g\|_2 $ to control paraproduct terms and derive quantitative decay.

Experimental results

Research questions

• RQ1 Can the Fourier-analytic method of Bourgain be extended to detect non-linear patterns $ \{x - t, x - P(t)\} $ in sets of positive upper density?
• RQ2 What is the quantitative lower bound on the density of such configurations when $ P(t) $ is a non-flat curve or polynomial without linear/constant terms?
• RQ3 How does the presence of curvature or non-flatness affect the existence and density of such patterns in dense sets?
• RQ4 Can the lower bound on the bilinear form be improved beyond previous results using refined frequency decomposition and paraproduct estimates?
• RQ5 What is the dependence of the lower bound on the polynomial’s coefficients and the set’s density $ \delta $?

Key findings

• For any set $ A \subset \mathbb{R} $ of positive upper density $ \delta $, there exists $ x_R \in A $ such that for all large $ R $, the set of $ t \in [0,R] $ with $ x_R - t, x_R - P(t) \in A $ has density at least $ c_P \delta^2 $, where $ c_P > 0 $ depends only on $ P $.
• The main result is quantitatively improved via Corollary 1.8, which gives a lower bound of order $ \frac{c_{\epsilon,P}}{2^{\delta^{-5-\epsilon}}} \cdot c_P \delta^3 $ for the infimum of the bilinear form over $ r \in (0,1] $.
• A key proposition shows that if the bilinear form is small on average, then the $ L^2 $-norm of the Fourier transform of $ \mathbf{1}_A $ must be large over certain frequency annuli, implying structured Fourier support.
• The proof establishes that the failure of the bilinear form to be large forces a large $ L^2 $-mass in the Fourier side, which contradicts the smallness assumption unless the set has a certain frequency structure.
• The result holds for a broad class of non-flat curves, including real analytic functions vanishing to order at least 2 at 0, Laurent polynomials with degrees $ \geq 2 $, and functions like $ |t|^\alpha |\log|t||^\beta $ with $ \alpha \neq 0,1 $.
• The dependence of constants $ c_P, C_P $ on $ \|P\| $ (the $ \ell^1 $-sum of coefficients) is explicit, and the bounds are robust under rescaling and dyadic decomposition.