Additive decompositions induced by multiplicative characters over finite fields

TL;DR

This paper generalizes additive properties of quadratic residues in finite fields to multiplicative quadratic and cubic characters, deriving representation counts and exploring related additive subset problems.

Contribution

It introduces a new generalization of additive properties for multiplicative characters over finite fields, extending Perron's classical results.

Findings

Derived formulas for the number of representations of elements as sums of coset members.

Established a connection between these representations and polynomial characteristic functions.

Explored a quasi-duality related to additive subset problems within cosets.

Abstract

In 1952, Perron showed that quadratic residues in a field of prime order satisfy certain ad- ditive properties. This result has been generalized in different directions, and our contribution is to provide a further generalization concerning multiplicative quadratic and cubic characters over any finite field. In particular, recalling that a character partitions the multiplicative group of the field into cosets with respect to its kernel, we will derive the number of representations of an element as a sum of two elements belonging to two given cosets. These numbers are then related to the equations satisfied by the polynomial characteristic functions of the cosets. Further, we show a connection, a quasi-duality, with the problem of determining how many elements can be added to each element of a subset of a coset in such a way as to obtain elements still belonging to a subset of a coset.