Restriction Algebras of Fourier–Stieltjes Transforms of Radon Measures

Abstract

If $\mu$ is an arbitrary bounded Radon measure $\mu$ on ${\mathbb {R}}^n,$ we denote by ${{\widehat{\mu }}}$ the Fourier–Stieltjes transform of $\mu$ and by $\sigma$ the pure point part of $\mu .$ A closed $\varLambda \subset {{\mathbb {R}}}^n$ is a gregarious set if the following property is satisfied:

$$\begin{aligned} (\forall \mu )\, {{\widehat{\mu }}}=0\quad \mathrm{on}\, \varLambda \Rightarrow {{\widehat{\sigma }}}=0\, \mathrm{on}\,\varLambda . \end{aligned}$$

Gregarious sets are studied in this essay.