发表机构
University of Chinese Academy of Sciences（中国科学院大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过过渡原理和关联几何，证明了若干结构化序列（如阶乘、Pochhammer乘积等）在素域上的值集具有多项式下界，并推广到Möbius变换情形，得到超过平方根尺度的结果。
AI 中文摘要
我们建立了两个过渡原理，用于对素域上由结构化序列生成的值集给出下界。一个具有$M$次内部过渡且商重数有界的倒数仿射族，其像集大小$\ngg \nin{M,p}^{8/15}$。这恢复了阶乘剩余界，并为算术Pochhammer乘积、高斯$q$-阶乘、错排数和有序子集数给出了相同的指数。第二个定理处理连续比值在非退化Möbius变换下演化的非零序列：其值集大小$\ngg \nin{M,p}^{1/2+\neta}$，其中$\neta>0$为绝对常数。作为推论，Pascal三角形的固定行以及Catalan序列和中心二项式序列的初始半块超过了平方根尺度。证明分别结合了过渡商与笛卡尔积点线关联几何，以及$\nathrm{SL}_2(\nathbb{F}_p)$中基于Bourgain扩张的关联定理。
英文摘要
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 $\gg \min{M,p}^{8/15}$. 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 $\gg \min{M,p}^{1/2+η}$ for an absolute constant $η>0$. 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 $\mathrm{SL}_2(\mathbb{F}_p)$.
Comments20 pages, no figures. Comments are welcome. A partial Lean 4 formalization is available at https://github.com/hxypqr/finite-field-value-sets
