2017/02/28 by Thomas Brendan Murphy, Brendan Murphy, Giorgis Petridis +4 · 54 citations
Computer Science · Mathematics · #Advanced Graph Theory Research #Cardinality (data modeling) #Combinatorics #Discrete mathematics #Geometry #Graph theory and applications #Limits and Structures in Graph Theory #Mathematics #Multiplication (music) #Product (mathematics) #Set (abstract data type) #Type (biology) #math.CO #math.NT #msc:11B30
paper · pdf · doi:10.1112/s0025579319000044
published in Mathematika 65(3), 588-642 (Wiley) · 59 pages. Added Theorem 2 and improved Theorem 1. This version is different to the one in print
crossref issued 2019/01/01 · crossref published 2019/01/01 · crossref published-print 2019/01/01 · openalex publication_date 2019/01/01 · crossref published-online 2019/04/02 · crossref created 2019/04/02 · arxiv created 2019/05/19 · arxiv updated 2019/05/22 · openalex created_date 2020/11/23 · crossref deposited 2021/02/09 · crossref indexed 2026/07/31 · openalex updated_date 2026/08/05
We prove a range of new sum-product type growth estimates over a general field , in particular the special case . They are unified by the theme of “breaking the threshold”, epitomizing the previous state of the art. This concerns two questions pivotal for the sum-product theory, which are lower bounds for the number of distinct cross-ratios determined by a finite subset of , as well as the number of values of the symplectic form determined by a finite subset of . We establish the estimate for cardinality of the set of distinct cross-ratios, defined by triples of elements of a set (sufficiently small if has positive characteristic, similarly for the rest of the estimates), pinned at infinity. The cross-ratio bound enables us to break the threshold in the second question: for a non-collinear point set , the number of distinct values of the symplectic form on pairs of distinct points of is , with an explicit . Symmetries of the cross-ratio underlie its local growth properties under both addition and multiplication, yielding an onset of growth of products of difference sets, which is another main result herein. Our proofs make use of specially suited applications of new incidence bounds over , which apply to higher moments of representation functions. The technical thrust of the paper uses additive combinatorics to relate and adapt these higher moment bounds to growth estimates. A particular instance of this is breaking the threshold in the few sums, many products question over any , by showing that if is sufficiently small and has additive doubling constant , then . This result has a second moment version, which allows for new upper bounds for the number of collinear point triples in the set , the quantity often arising in applications of geometric incidence estimates.