• Corpus ID: 244799354

A multi-linear geometric estimate

@inproceedings{Aten2021AMG,
  title={A multi-linear geometric estimate},
  author={Charlotte Aten and Alex Iosevich},
  year={2021}
}

• C. Aten, A. Iosevich
• Published 1 December 2021
• Mathematics

Abstract. 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 ⊂ Fq is sufficiently large and ̟ is a non-degenerate multi-linear form then ̟ 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 F∗q belongs to A ·A ·A+ A · A · A ·A ·A ·A where A is a union of cosets of a subgroup of F∗q .