Lower bounds for some value sets over finite fields: incidence geometry and Bourgain's group expansion theorem

We develop two transition principles for lower-bounding value sets generated by structured sequences over prime fields. A reciprocal-affine family with $M$ internal transitions and bounded quotient multiplicity has image size $\gg \min {M,p}^{8\mathrm{/}15}$. This recovers the factorial-residue bound and yields the same exponent for arithmetic Pochhammer products, Gaussian $q$-factorials, derangement numbers, and the numbers of ordered subsets. A second theorem treats nonzero sequences whose consecutive ratios evolve under a nondegenerate M"obius transformation: their value sets have size $\gg \min {M,p}^{1\mathrm{/}2+\eta }$ for an absolute constant $\eta >0$. As consequences, fixed rows of Pascal’s triangle and the initial half-blocks of the Catalan and central binomial sequences exceed the square-root scale. The proofs combine transition quotients with, respectively, Cartesian-product point-line incidence geometry and Bourgain’s expansion-based incidence theorem in ${\mathrm{S}\mathrm{L}}_2({\mathbb{F}}_p)$.