The Periodic Dilation Completeness Problem: Cyclic vectors in the Hardy space over the infinite-dimensional polydisk

The classical completeness problem raised by Beurling and independently by Wintner asks for which $\psi\in L^2(0,1)$, the dilation system $\{\psi(kx):k=1,2,\cdots\}$ is complete in $L^2(0,1)$, where $\psi$ is identified with its extension to an odd $2$-periodic function on $\mathbb{R}$. This difficult problem is nowadays commonly called as the Periodic Dilation Completeness Problem (PDCP). By Beurling's idea and an application of the Bohr transform, the PDCP is translated as an equivalent problem of characterizing cyclic vectors in the Hardy space $\mathbf{H}_\infty^2$ over the infinite-dimensional polydisk for coordinate multiplication operators. In this paper, we obtain lots of new results on cyclic vectors in the Hardy space $\mathbf{H}_\infty^2$. In almost all interesting cases, we obtain sufficient and necessary criterions for characterizing cyclic vectors, and hence in these cases we completely solve the PDCP. Our results cover almost all previous known results on this subject.