[Paper Review] An application of the sum-product phenomenon to sets having no solutions of several linear equations

Ilya D. Shkredov|arXiv (Cornell University)|Sep 21, 2016

Limits and Structures in Graph Theory16 references3 citations

TL;DR

This paper applies the sum-product phenomenon in finite fields to establish bounds on the size of subsets of $\mathbb{F}_p$ that avoid multiple linear equations of the form $x_1 + a_jx_2 + b_jx_3 = b_j$. Using spectral methods and incidence theory, it proves that such sets have size $O(p / t^{\kappa})$ for $\kappa < 3/20$, improving upon known bounds and showing that the exponent cannot exceed $1/3$ without new techniques. The result has applications to non-averaging sets, collinear triples, and mixed energies.

ABSTRACT

We prove that for an arbitrary $κ\le \frac{1}{3}$ any subset of $\mathbf{F}_p$ avoiding $t$ linear equations with three variables has size less than $O(p/t^κ)$. We also find several applications to problems about so--called non--averaging sets, number of collinear triples and mixed energies.

Motivation & Objective

• To determine the maximal size of subsets in $\mathbb{F}_p$ that avoid $t$ linear equations of the form $x_1 + a_jx_2 + b_jx_3 = b_j$ with nonzero coefficients.
• To investigate how the sum-product phenomenon constrains the size of such sets, especially when $t \to \infty$ and $p \to \infty$.
• To establish quantitative upper and lower bounds on the size of sets avoiding $t$ such equations, with explicit exponents in $t$.
• To apply the results to problems in additive combinatorics, including non-averaging sets, collinear triples in Cartesian products, and mixed energy estimates.

Proposed method

• Uses the spectrum of a set $A \subseteq \mathbb{F}_p$, defined via large Fourier coefficients, to analyze its additive and multiplicative structure.
• Applies precise incidence theorems from [9] and their extensions from [1, 7, 10] to control the number of solutions to linear equations.
• Employs the dual set (spectrum) to extend incidence results to large sets $A$, not just small ones, via spectral energy bounds.
• Establishes that the spectrum of $A$ has small multiplicative energy, and contains a large subset with even smaller multiplicative energy.
• Uses the Cauchy–Schwarz inequality and pigeonholing on Fourier coefficients to bound the deviation of mixed energy sums from their expected value.
• Derives a key estimate involving $\|\widehat{A}\|'_{\infty}$, the sup-norm of the Fourier transform restricted to large coefficients, to control error terms in energy estimates.

Experimental results

Research questions

• RQ1 Can the sum-product phenomenon be used to bound the size of sets avoiding multiple linear equations in $\mathbb{F}_p$?
• RQ2 What is the best possible exponent $\kappa$ such that $|A| = O(p / t^{\kappa})$ for sets avoiding $t$ equations of the form $x_1 + a_jx_2 + b_jx_3 = b_j$?
• RQ3 Is it possible to improve the exponent beyond $\kappa = 1/3$, and what would that imply for additive combinatorics?
• RQ4 How do spectral properties of $A$, such as the size of its spectrum and its multiplicative energy, relate to its avoidance of linear equations?
• RQ5 Can the method be extended to estimate mixed energy $\sum_{x \in X} \mathsf{E}^+(A, xA)$ for large sets $A$?

Key findings

• For any $\kappa < 3/20$, if $A \subseteq \mathbb{F}_p$ avoids $t$ equations of the form $x_1 + a_jx_2 + b_jx_3 = b_j$ with distinct ratios or unique coordinates, then $|A| = O(p / t^{\kappa})$.
• The paper constructs a set avoiding $t$ such equations with size $\gg p / t^{1/2}$, showing that the exponent $1/2$ is tight in some cases.
• The spectrum of any set $A \subseteq \mathbb{F}_p$ has small multiplicative energy, and contains a large subset with even smaller multiplicative energy.
• For $|A| \gtrsim p^{11/14}$, the set $R[A]$ of ratios of differences has size at least $ (1 - o(1))p $, implying near-uniform distribution of ratios.
• A new estimate for mixed energy is derived: $\left| \sum_{x \in X} \mathsf{E}^+(A, xA) - \frac{|X||A|^4}{p} \right| \lesssim \delta^{8/3} |X|^{1/2} p^3 \left( \frac{\|\widehat{A}\|'_{\infty}}{|A|} \right)^{2/3} $, valid under a condition on $|X|$.
• The bound on $|A|$ cannot be improved beyond $O(p / t^{1/3})$ without new ideas, as $1/3$ is believed to be the natural limit of current techniques.