[Paper Review] Additive energy of regular measures in one and higher dimensions, and the fractal uncertainty principle

Laura Cladek, Terence Tao|arXiv (Cornell University)|Dec 4, 2020

Limits and Structures in Graph Theory20 references4 citations

TL;DR

This paper establishes improved bounds on the additive energy of Ahlfors–David regular measures in one and higher dimensions, leveraging geometric and harmonic analysis techniques to achieve polynomial dependence on the regularity constant $ C $, significantly improving over prior quasipolynomial bounds. These results yield new cases of the fractal uncertainty principle in odd dimensions, particularly for non-integer dimensions.

ABSTRACT

We obtain new bounds on the additive energy of (Ahlfors-David type) regular measures in both one and higher dimensions, which implies expansion results for sums and products of the associated regular sets, as well as more general nonlinear functions of these sets. As a corollary of the higher-dimensional results we obtain some new cases of the fractal uncertainty principle in odd dimensions.

Motivation & Objective

• To improve the exponent in additive energy bounds for one-dimensional regular measures, replacing quasipolynomial dependence on the regularity constant $ C $ with polynomial decay.
• To extend these improved bounds to higher-dimensional regular measures and apply them to nonlinear functions and sum-product phenomena.
• To establish new cases of the fractal uncertainty principle in odd dimensions using the refined additive energy estimates.
• To demonstrate that the improved bounds arise from geometric structure and dyadic decomposition in the real line, leveraging order and regularity.
• To show that the improved bounds lead to stronger $ L^p $-restriction estimates and stronger uncertainty principles in fractal settings.

Proposed method

• Use of dyadic (K-adic) interval decomposition to exploit the order structure of $ \mathbb{R} $, enabling finer control over additive energy at small scales.
• Application of the Gowers–Cauchy–Schwarz inequality to bound the $ U^2 $-norm of characteristic functions, linking it to additive energy via the $ \|1_A * 1_B\|_{L^2}^2 $-norm.
• Rescaling of sets to normalize to unit scale, transforming the problem into a regularity condition on rescaled measures with controlled $ C^{O(1)} $-regularity constants.
• Use of the $ \delta $-regularity condition to control measure of balls at scale $ r $, ensuring $ \mu(B(x,r)) \sim r^\delta $, and applying volume-packing arguments.
• Application of the fractal uncertainty principle via duality and restriction theory, connecting $ L^4 $-bounds on Fourier transforms to additive energy via $ \|\mu * \nu\|_{U^2} $ estimates.
• Use of the $ C^2 $-smoothness of nonlinear functions $ F $ to linearize local behavior and reduce the problem to sumset estimates in $ \mathbb{R}^d $.

Experimental results

Research questions

• RQ1 Can the exponent $ \beta $ in the additive energy bound $ \mathcal{E}(\mu, r) \lesssim r^{\delta + \beta} $ be improved from quasipolynomial to polynomial in the regularity constant $ C $ in one dimension?
• RQ2 To what extent can the improved additive energy bounds in one dimension be extended to higher-dimensional regular measures and nonlinear functions?
• RQ3 What new cases of the fractal uncertainty principle can be established using the refined additive energy estimates in odd dimensions?
• RQ4 How does the order structure of $ \mathbb{R} $ enable stronger bounds than general additive combinatorics in non-integer dimensions?
• RQ5 Can the improved bounds on additive energy lead to stronger $ L^p $-restriction estimates for fractal measures?

Key findings

• The paper achieves a polynomial improvement in the additive energy exponent, showing $ \beta = c \min(\delta, 1 - \delta) C^{-25} $ for some absolute $ c > 0 $, significantly improving over the quasipolynomial $ \beta \sim \delta \exp(-K(1 - \delta)^{-14}(1 + \log^{14} C)) $ from Dyatlov and Zahl.
• The improved bound is established using an elementary argument based on dyadic decomposition and the order structure of $ \mathbb{R} $, avoiding deep inverse theorems.
• The higher-dimensional extension of the additive energy bound leads to new cases of the fractal uncertainty principle in odd dimensions, particularly for non-integer $ \delta $.
• The bound $ \|\mathcal{F}_h 1_{Y_h}\|_{L^4} \lesssim h^{d/2 - 3\delta/4 + 2\beta} $ is established via $ \|\mu_Y * \nu_{B(0,2h)}\|_{U^2} \lesssim h^{d/2 - \delta} \mathcal{E}(\mu_Y, h)^{1/4} $, with $ \mathcal{E}(\mu_Y, h) \lesssim h^{\delta + \beta} $.
• The results are robust under rescaling and apply to $ C^2 $-smooth nonlinear functions $ F $, with the key step being the linearization of $ F $ near a point and reduction to sumset estimates.
• The method applies to sets $ Y_h $ with measure $ \gtrsim r^{d - \delta} $, relaxing strict regularity assumptions on $ Y $, and allows for generalization beyond regular measures.