Approximate homomorphisms between the boolean cube and groups of prime order

The purpose of this note is to highlight a question raised by Shachar Lovett [Lov], and to offer some motivation for its study. Our interest is in the existence of injections f : pZ{2Zqn Ñ Z{pZ (where p is an odd prime) that are ‘approximately homomorphisms’ in the sense that f px` yq “ f pxq ` f pyq for many x and y. Lovett noted that if p ą 2n then the map f : pZ{2Zqn Ñ Z{pZ that ‘embeds in binary’ is an injection with1