We inspect the relationship between relative Fourier multipliers on noncommutative Lebesgue–Orlicz spaces of a discrete group $\Gamma$ and relative Toeplitz-Schur multipliers on Schatten–von-Neumann–Orlicz classes. Four applications are given: lacunary sets, unconditional Schauder bases for the subspace of a Lebesgue space determined by a given spectrum $\Lambda \,\subseteq \,\Gamma$, the norm of the Hilbert transformand the Riesz projection on Schatten–von-Neumann classes with exponent a power of 2, and the norm of Toeplitz Schur multipliers on Schatten–von-Neumann classes with exponent less than 1.

We consider maximal operators in the plane, defined by Cantor sets of directions, and show such operators are not bounded on ${{L}^{2}}$ if the Cantor set has positive Hausdorff dimension.

A weighted generalization of a p-Sidon set, called an (a, p)-Sidon set, is introduced and studied for infinite, non-abelian, connected, compact groups G. The entire dual object Ĝ is shown never to be central (p − 1, p)-Sidon for 1 ≦ p < 2, nor central (1 + ε, 2)-Sidon for ε > 0. Local (p, p)-Sidon sets are shown to be identical to local Sidon sets. Examples are constructed of infinite non-Sidon sets which are central (2p − 1, p)-Sidon, or (p − 1, p)-Sidon, for 1 < p < 2. Full m-fold FTR sets are proved not to be central (a, 2m/(m + 1))-Sidon for any a > 1.

Let G be any compact group, connected or disconnected, with dual object Ĝ. We define a family of local Sidon subsets of Ĝ in terms of allowable images of the representations. Using this family we develop a straightforward criterion whereby the existence of infinite Sidon subsets of Ĝ may be decided.