Lacunary sets based on Lorentz spaces

A new lacunary set for compact abelian groups is introduced; this is called a $\Lambda(p,q)$ set. This set is defined in terms of the Lorentz spaces and is shown to be a generalization of $\Lambda(p)$ sets and Sidon sets. A number of functional-analytic statements about $\Lambda(p,q)$ sets are established by making use of the structural similarities between $L^{p}$ spaces and Lorentz spaces. These statements are analogous to several well-known properties of a set which are equivalent to the definition of a $\Lambda(p)$ set. Some general set-theoretic and arithmetic properties of $\Lambda(p,q)$ sets are also developed; these properties extend known results on the structure of $\Lambda(p)$ sets. Open problems and directions for further research are outlined.