Rajchman multiplication and the Zygmund sets \(U(\varepsilon)\) (Q2773419)

A. Zygmund introduced a generalized notion of a set of uniqueness for Fourier series, in which one stipulates that \(E\) is a \(U(\varepsilon)\) set provided the conditions that the trigonometric series \(\sum_n a_n e^{2\pi i nt}\) converges to zero outside of \(E\) while \(| a_n| \leq \varepsilon_n\) together imply that \(\sum_n a_n e^{2\pi i nt}\) converges to zero identically. Convergence here may be taken in the sense of symmetric partial sums. Here \(\varepsilon_n\to 0\) as \(n\to\infty\). Kahane and Katznelson proved in 1973 the existence of \(U(\varepsilon)\) sets in the torus having full measure. In the present paper the author provides a simpler construction of such sets, based on Rajchman's multiplication, which assigns a meaning to the product of two formal power series \(S_i=\sum_{n\in\mathbb{Z}} a_n^{(i)}z^n\) whenever the product of their comoduli \(| S| _i\equiv \sum_n | a_n| z^n\) is well-defined. The author also provides interesting historical commentary regarding the relationship between multiplication and uniqueness.