2015/12/21 by Esen Aksoy Yazici, Brendan Murphy, Yazici, Esen Aksoy +6
Mathematics · #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory #Mathematical Approximation and Integration #math.CO
paper · pdf · doi:10.48550/arxiv.1512.06613
24pp
arxiv created 2015/12/21 · openalex publication_date 2015/12/21 · arxiv updated 2015/12/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let F be a field of characteristic p>2 and A⊂ F have sufficiently small cardinality in terms of p. We improve the state of the art of a variety of sum-product type inequalities. In particular, we prove that |AA|2|A+A|3 ≫ |A|6, |A(A+A)|≫ |A|3/2. We also prove several two-variable extractor estimates: |A(A+1)| ≫|A|9/8, |A+A2|≫ |A|11/10, |A+A3|≫ |A|29/28, |A+1/A|≫ |A|31/30. Besides, we address questions of cardinalities |A+A| vs |f(A)+f(A)|, for a polynomial f, where we establish the inequalities max(|A+A|, |A2+A2|)≫ |A|8/7, max(|A-A|, |A3+A3|)≫ |A|17/16. Szemerédi-Trotter type implications of the arithmetic estimates in question are that a Cartesian product point set P=A× B in F2, of n elements, with |B|≤ |A|< p2/3 makes O(n3/4m2/3 + m + n) incidences with any set of m lines. In particular, when |A|=|B|, there are ≪ n9/4 collinear triples of points in P, ≫ n3/2 distinct lines between pairs of its points, in ≫ n3/4 distinct directions. Besides, P=A× A determines ≫ n9/16 distinct pair-wise distances. These estimates are obtained on the basis of a new plane geometry interpretation of the incidence theorem between points and planes in three dimensions, which we call collisions of images.