Spectral properties of dynamical tensor powers, and tensor factorizations of simple Lebesgue spectrum

The paper fully proves the tensor factorization (Theorem 1) and is correct there, but for the symmetric powers (Theorem 2) it states a result for all n>1 while only giving a detailed construction and argument for n=2, and even then it verifies absolute continuity rather than equivalence to Lebesgue except under an extra rate condition. The model’s Part (1) conclusion is correct but misattributes the method (it invokes ‘rank-one/Sidon’ for U where the paper uses a Bernoulli/Cantor construction). In Part (2) the model conflates ‘Lebesgue spectrum’ with ‘absolutely continuous’ and relies on the paper without supplying missing general-n details. Hence both are incomplete for the second claim, and the model has some methodological misstatements.