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The classical completeness problem raised by Beurling and independently byWintner 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 $\psi$ isidentified with its extension to an odd $2$-periodic function on $\mathbb{R}$.This difficult problem is nowadays commonly called as the Periodic DilationCompleteness Problem (PDCP). By Beurling's idea and an application of the Bohrtransform, the PDCP is translated as an equivalent problem of characterizingcyclic vectors in the Hardy space $\mathbf{H}_\infty^2$ over theinfinite-dimensional polydisk for coordinate multiplication operators. In thispaper, we obtain lots of new results on cyclic vectors in the Hardy space$\mathbf{H}_\infty^2$. In almost all interesting cases, we obtain sufficientand necessary criterions for characterizing cyclic vectors, and hence in thesecases we completely solve the PDCP. Our results cover almost all previous knownresults on this subject.

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