Equidistribution of orbits at polynomial times in rigid dynamical systems

We study distribution of orbits sampled at polynomial times for uniquely ergodic topological dynamical systems $(X,T)$. First, we prove that if there exists an increasing sequence $(q_n)$ for which the rigidity condition $$\underset{t<q_{n+1}^{4\mathrm{/}5}}{\max }\underset{x\in X}{\sup }d(x,T^{t{q}_n}x)=o(1)$$ is satisfied, then all square orbits $(T^{n^2}x)$ are equidistributed (with respect to the only invariant measure). We show that this rigidity condition might hold for weakly mixing systems, and so as a consequence we obtain first examples of weakly mixing systems where such an equidistribution holds. We also show that for integers $C>1$ a much weaker rigidity condition $$\underset{t<q_n^{C-1}}{\max }\underset{x\in X}{\sup }d(x,T^{t{q}_n}x)=o(1)$$ implies density of all orbits $(T^{n^C}x)$ in totally uniquely ergodic systems, as long as the sequence $(\omega (q_n))$ is bounded.