2 problems

Let $(X,\mu,T)$ be an ergodic system satisfying the spectral hypothesis of Corollary, and let $(Y,\nu,S)$ be an ergodic nilsystem. A joining of these systems is a joining of the me…

In the setting of shifted packets $P_D^{\mathrm{shifted}}$ on $X_1\times\cdots\times X_r$, assume individual equidistribution, or assume $k_1=\dots=k_r=1$. For every pair $i\ne j$…