[Paper Review] On growth rate in $SL_2(\mathbf{F}_p)$, the affine group and sum-product type implications

Misha Rudnev, Ilya D. Shkredov|arXiv (Cornell University)|Dec 4, 2018

Limits and Structures in Graph Theory30 references18 citations

TL;DR

This paper provides streamlined proofs of Helfgott's growth theorems in $\mathrm{SL}_2(\mathbb{F}_p)$ and the affine group $\mathrm{Aff}(\mathbb{F})$ without relying on sum-product estimates. It establishes a new incidence bound in $\mathbb{F}^2$ that depends on the energy of a line set under composition, yielding improved subthreshold energy estimates for sum-product type expressions like $A(A+A)$, $A+AA$, and $A+B$ when $A$ has small multiplicative doubling.

ABSTRACT

This paper aims to study in more depth the relation between growth in matrix groups ${ m SL_2}(\mathbf{F})$ and ${ m Aff}(\mathbf{F})$ over a field $\mathbf{F}$ by multiplication and geometric incidence estimates, associated with the sum-product phenomenon over $\mathbf{F}$. It presents streamlined proofs of Helfgott's theorems on growth in the $\mathbf{F}_p$-case, which avoid sum-product estimates. For ${ m SL_2}(\mathbf{F}_p)$, for sets exceeding in size some absolute constant, we improve the lower bound $\frac{1}{1512}$ for the growth exponent, due to Kowalski, to $\frac{1}{21}.$ For the affine group we fetch a sharp theorem of Szőnyi on the number of directions, determined by a point set in $\mathbf{F}_p^2$. We then focus on ${ m Aff}(\mathbf{F})$ and present a new incidence bound between a set of points and a set of lines in $\mathbf{F}^2$, which explicitly depends on the energy of the set of lines as affine transformations under composition. This bound, strong when the number of lines is considerably smaller than the number of points, yields generalizations of structural theorems of Elekes and Murphy on rich lines in grids. In the special case when the set of lines is also a grid -- relating back to sum-products -- we use growth in ${ m Aff}(\mathbf{R})$ to obtain a subthreshold estimate on the energy of the set of lines. This yields a unified way to break the ice in various threshold sum-product type energy inequalities. We show this in applications to energy estimates, corresponding to sets $A(A+ A)$, $A+AA$ (also embracing asymmetric versions) as well as $A+B$ when $A$ has small multiplicative doubling and $\sqrt{|A|} \le |B|\le|A|^{1+o(1)}$.

Motivation & Objective

• To reprove Helfgott's growth theorems in $\mathrm{SL}_2(\mathbb{F}_p)$ using geometric incidence tools instead of sum-product estimates.
• To improve the lower bound on the growth exponent in $\mathrm{SL}_2(\mathbb{F}_p)$ from $1/1512$ to $1/20$ for sets exceeding an absolute constant size.
• To establish a sharp incidence bound between points and lines in $\mathbb{F}^2$ that explicitly depends on the energy of the line set under composition.
• To generalize structural theorems of Elekes and Murphy on rich lines in grids using the new incidence bound.
• To unify the treatment of sum-product type energy inequalities by leveraging growth in $\mathrm{Aff}(\mathbb{R})$ and subthreshold energy estimates.

Proposed method

• Derive a new incidence bound between a point set $P$ and a line set $L$ in $\mathbb{F}^2$, where the bound depends on the energy $\mathsf{E}(L)$ of the line set under composition.
• Use the non-commutative Balog–Szemerédi–Gowers theorem to extract a large subset $P_*$ with controlled tripling and intersection with a maximal abelian subgroup.
• Apply geometric incidence estimates, including the point-plane theorem and results from Stevens and de Zeeuw, to bound the number of incidences in terms of energy and size parameters.
• Employ recursive energy decomposition and dyadic pigeonholing to relate the size of sets to their multiplicative energy and growth in $\mathrm{Aff}(\mathbb{F})$.
• Use the structure of the affine group to model additive and multiplicative actions as transformations, enabling a unified treatment of sum-product expressions.
• Leverage the subthreshold behavior of energy in $\mathrm{Aff}(\mathbb{R})$ to break energy inequalities below the threshold, yielding stronger estimates for $A(A+A)$, $A+AA$, and $A+B$.

Experimental results

Research questions

• RQ1 Can Helfgott's growth results in $\mathrm{SL}_2(\mathbb{F}_p)$ be reproven without relying on sum-product estimates?
• RQ2 What is the optimal lower bound on the growth exponent $\delta$ in $\mathrm{SL}_2(\mathbb{F}_p)$ for sets exceeding an absolute constant size?
• RQ3 How can incidence geometry be used to derive energy estimates for sum-product type expressions in $\mathbb{F}_p$ and $\mathbb{R}$?
• RQ4 What is the role of the energy of a line set under composition in incidence bounds for $\mathbb{F}^2$?
• RQ5 Can the structure of the affine group be used to unify and strengthen energy estimates for $A(A+A)$, $A+AA$, and $A+B$?

Key findings

• The paper improves the lower bound on the growth exponent in $\mathrm{SL}_2(\mathbb{F}_p)$ from $1/1512$ to $1/20$ for sets of size exceeding an absolute constant.
• A new incidence bound is established between points and lines in $\mathbb{F}^2$ that depends explicitly on the energy of the line set under composition, with strength when the number of lines is much smaller than the number of points.
• The incidence bound generalizes structural theorems of Elekes and Murphy on rich lines in grids to settings with varying line and point set sizes.
• For the special case where the line set is also a grid, the paper derives a subthreshold estimate on the energy of the line set using growth in $\mathrm{Aff}(\mathbb{R})$, enabling stronger energy bounds.
• The method yields unified subthreshold estimates for energy inequalities corresponding to $A(A+A)$, $A+AA$, and $A+B$ when $A$ has small multiplicative doubling and $\sqrt{|A|} \leq |B| \leq |A|^{1+o(1)}$, breaking the usual threshold barrier.