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

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