Tame measures and Raikov systems on the circle (Q802101)

A probability measure \(\mu\) on the circle group \({\mathbb{T}}\) is tame if for each generalized character \(\phi\) of M(\({\mathbb{T}})\) there exists \(\gamma\in {\mathbb{T}}\) and \(a\in {\mathbb{C}}\) such that \(\phi_{\mu}=a.\gamma\) \(\mu\) a.e. [cf. \textit{G. Brown} and \textit{W. Moran}, Acta Math. 132, 77-109 (1974; Zbl 0278.43004)]. The paper considers the relationship between tameness of a probability measure \(\mu\) and the size of sum sets of positive \(\mu\)-measure. Specifically, the author constructs a tame probability measure \(\mu\) on \({\mathbb{T}}\) which is concentrated on a set E such that \(E+E+...+E\) (n times) has empty interior for all positive integers n. The construction of \(\mu\) involves techniques similar to those developed by \textit{Th. W. Körner} [Ann. Inst. Fourier 20, 219-324 (1970; Zbl 0196.084)] to exhibit a measure in \(M_ 0({\mathbb{T}})\) which has an independent set as support. Discrete approximates to such measures are procuded and an infinite convolution of these gives the required measure.