On the growth rate in SL₂(F_p), the affine group and sum-product type implications

This paper aims to study in more depth the relation between growth in matrix groups (Formula presented.) and (Formula presented.) over a field (Formula presented.) by multiplication and geometric incidence estimates, associated with the sum-product phenomenon over (Formula presented.). It presents streamlined proofs of Helfgott's theorems on growth in the (Formula presented.) -case, which avoid sum-product estimates. For (Formula presented.), for sets exceeding in size some absolute constant, we improve the lower bound (Formula presented.) for the growth exponent, due to Kowalski, to (Formula presented.) and establish an upper bound (Formula presented.). for the Cayley graph diameter exponent. For the affine group, we fetch a sharp theorem of Szőnyi on the number of directions, determined by a point set in (Formula presented.). We then focus on (Formula presented.) and present a new incidence bound between a set of points and a set of lines in (Formula presented.), 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 generalisations 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 (Formula presented.) 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 (Formula presented.), (Formula presented.) (also embracing asymmetric versions) as well as (Formula presented.) when A has small multiplicative doubling and (Formula presented.).