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

Abstract

This paper aims to study in more depth the relation between growth in matrix groups ${{\rm SL_2}}(\mathbf{F})$ and ${{\rm Aff}}(\mathbf{F})$ 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 ${{\rm SL_2}}(\mathbf{F}_p)$, 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 Sz\H{o}nyi on the number of directions, determined by a point set in $\mathbf{F}_p^2$. We then focus on ${{\rm Aff}}(\mathbf{F})$ 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 ${{\rm Aff}}(\mathbf{R})$ 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 $\sqrt{|A|} \le |B|\le|A|^{1+o(1)}$.