Lower bounds for some value sets over finite fields: incidence geometry and Bourgain's group expansion theorem
Abstract
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 ≫ minM,p8/15. 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 ≫ minM,p1/2+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 SL2(𝔽p).
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BibTeX
@misc{hu2026lower,
title = {{Lower bounds for some value sets over finite fields: incidence geometry and Bourgain's group expansion theorem}},
author = {Xiyu Hu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.05652},
eprint = {2609.05652},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.05652},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Lower bounds for some value sets over finite fields: incidence geometry and Bourgain's group expansion theorem
AU - Xiyu Hu
PY - 2026
DA - 2026/09/04
PB - arXiv
DO - 10.48550/arxiv.2609.05652
UR - https://arxiv.org/abs/2609.05652
AB - 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 ≫ minM,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 ≫ minM,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 SL_2(𝔽_p).
ER -