An ergodic automorphism $\bf T$ with singular spectrum of $\bf T^{\otimes n}$ and Lebesgue one of $\bf T^{\otimes (n+1)}$

Abstract

For any natural $n$, and real $\alpha\geq 0$ we construct an ergodic automorphism $T$ such that its tensor powers $T^{\otimes n}$ have singular spectra if $n\leq 1+\alpha /2$, and Lebesgue if $n\, > 1+\alpha/2$.