Equidistribution of orbits at polynomial times in rigid dynamical systems

We study the distribution of orbits sampled at polynomial times for uniquely ergodic topological dynamical systems ( X , T ) (X, T) . First, we prove that if there exists an increasing sequence ( q n ) (q_n) for which the rigidity condition \[ max t > q n + 1 4 / 5 sup x ∈ X d ( x , T t q n x ) = o ( 1 ) \max _{t>q_{n+1}^{4/5}}\sup _{x\in X}d(x, T^{tq_n}x)=o(1) \] is satisfied, then, for any integer polynomial P P of degree 2, all orbits ( T P ( n ) x ) (T^{P(n)}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 C>1 a much weaker rigidity condition \[ max t > q n C − 1