  We study the interplay between notions of quasirandomness for additive sets
and for hypergraphs. In particular, we show a strong connection between the
notions of Gowers uniformity in the additive setting and discrepancy-type
measures of quasirandomness in the hypergraph setting. Exploiting this
connection, we provide a long list of disparate quasirandom properties
regarding both additive sets and Cayley-type hypergraphs constructed from such
sets, and show that these properties are all equivalent (in the sense of Chung,
Graham and Wilson) with polynomial bounds on their interdependences.
