Moderate expanders over rings

Abstract

In this note, we provide a large class of moderate expanders with the exponents 38 and 513 over arbitrary finite fields and prime fields, respectively. Our main ingredients are an energy result due to the third, fourth, sixth listed authors and Shen (2019) and a theorem on two-variable expanding functions given by Hegyvári and Hennecart (2009). Using the same approach, we derive similar results in the setting of finite local and principal rings.