On the Wiener-Pitt phenomenon for algebra of Rajchman multipliers on Hardy space

We show that any positive Rajchman measure of Minkowski dimension $0$ has a non-natural spectrum as an element of the multiplier algebra of $H_0^1(\text{T})$. The proof is based on the estimation of the norm of the convolution operator given by a singular measure on $H_0^1(\text{T})$.