Sums of inverses in thin sets of finite fields

TL;DR

This paper establishes lower bounds for the size of sum-sets of reciprocals in finite fields and applies these results to improve bounds on multilinear Kloosterman sums, advancing understanding of additive structures in finite fields.

Contribution

It introduces new lower bounds for reciprocal sum-sets in finite fields and connects these bounds to improved estimates of multilinear Kloosterman sums.

Findings

Lower bounds for k-fold sum-sets of reciprocals in finite fields

Enhanced bounds for incomplete multilinear Kloosterman sums

New insights into additive structures in finite field extensions

Abstract

We obtain lower bounds for the cardinality of $k$-fold sum-sets of reciprocals of elements of suitable defined short intervals in high degree extensions of finite fields. Combining our results with bounds for multilinear character sums we obtain new results on incomplete multilinear Kloosterman sums in finite fields.