On growth rate in $S{L}_2({\mathbf{F}}_p)$, the affine group and sum-product type implications

Abstract

This paper aims to study in more depth the relation between growth in matrix groups $\mathrm{S}{\mathrm{L}}_2(\mathbf{F})$ and $\mathrm{Aff}(\mathbf{F})$ 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 $\mathrm{S}{\mathrm{L}}_2({\mathbf{F}}_p)$, 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 $\mathrm{Aff}(\mathbf{F})$ 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 $\mathrm{Aff}(\mathbf{R})$ 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 $\sqrt{|A|}\le |B|\le |A{|}^{1+o(1)}$.

43 pages

References in corpus (2)

• A sum-product estimate in fields of prime order
• Some remarks on the asymmetric sum--product phenomenon

Cited by in corpus (7)

• On the directions determined by Cartesian products and the clique number of generalized Paley graphs
• On sum sets of convex functions
• Incidences of Möbius transformations in ${\mathbb{F}}_p$
• An Energy Bound in the Affine Group
• Number of directions determined by a set in ${\mathbb{F}}_q^2$ and growth in $\mathrm{Aff}({\mathbb{F}}_q)$
• On multiplicative properties of combinatorial cubes
• Distribution of determinant of sum of matrices