The additive structure of the squares inside rings

TL;DR

This paper investigates the additive structure of squares within rings, analyzing sumsets in finite sets of squares, and extends the study to finite fields, providing estimates and computational results.

Contribution

It introduces new estimates for sumset sizes of squares in rings and finite fields, and computationally determines minimal sumset sizes for small finite fields.

Findings

Arithmetic progressions reduce sumset sizes efficiently

Provided bounds for sumsets in rings and finite fields

Computed minimal sumset sizes for small finite fields

Abstract

When defining the amount of additive structure on a set it is often convenient to consider certain sumsets; Calculating the cardinality of these sumsets can elucidate the set's underlying structure. We begin by investigating finite sets of perfect squares and associated sumsets. We reveal how arithmetic progressions efficiently reduce the cardinality of sumsets and provide estimates for the minimum size, taking advantage of the additive structure that arithmetic progressions provide. We then generalise the problem to arbitrary rings and achieve satisfactory estimates for the case of squares in finite fields of prime order. Finally, for sufficiently small finite fields we computationally calculate the minimum for all prime orders.