For several large classes of functions in the infinite polydisk Hardy space, a function is cyclic if and only if it is zero-free, yielding complete solutions to the Periodic Dilation Completeness Problem in those cases.
Beurling, The collected Works of Arne Beurling, 2: Harmonic Analysis , Contemp
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The Periodic Dilation Completeness Problem: Cyclic vectors in the Hardy space over the infinite-dimensional polydisk
For several large classes of functions in the infinite polydisk Hardy space, a function is cyclic if and only if it is zero-free, yielding complete solutions to the Periodic Dilation Completeness Problem in those cases.