Summable analogous to the Ivashev-Musatov Theorems

We investigate summable analogues of the classical Ivashev-Musatov Theorem and threshold phenomenons alike. In the setting of weighted $\ell^1$ and Orlicz sequence spaces, we exhibit elements with critically pathological support and range, both topologically and in the sense of measure theory. Our results complement earlier works of J. P. Kahane, Y. Katznelson and T. W. K\"orner.