Tensor Factorizations of a Unitary Operator with Simple Lebesgue Spectrum

We show that for all \(n,p>1\), there exists a unitary operator \(U\) such that the tensor product \(U\otimes U^p\otimes\dots\otimes U^{p^{n-1}}\) is a unitary operator with simple Lebesgue spectrum. Moreover, there exists an ergodic automorphism \(T\) such that the spectrum of \(T\odot T\) is simple, while the spectrum of \(T\otimes T\otimes T\) is absolutely continuous.