2021/12/01 by Charlotte Aten, Alex Iosevich, Aten, Charlotte +1
Mathematics · #11E20 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Finite Group Theory Research #Limits and Structures in Graph Theory #Mathematical Approximation and Integration #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2112.00810
openalex publication_date 2021/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We give a generalization of the geometric estimate used by Hart and the second author in their 2008 work on sums and products in finite fields. Their result concerned level sets of non-degenerate bilinear forms over finite fields, while in this work we prove that if E⊂\mathbbFqd is sufficiently large and \varpi is a non-degenerate multi-linear form then \varpi will attain all possible nonzero values as its arguments vary over E, under a certain quantitative assumption on the extent to which E is projective. We show that our bound is nontrivial in the case that n=3 and d=2 and construct examples of sets to which this applies. In particular, we give conditions under which every member of \mathbbFq^* belongs to A⋅ A⋅ A+A⋅ A⋅ A⋅ A⋅ A⋅ A where A is a union of cosets of a subgroup of \mathbbFq^*.