Almost-Everywhere Convergence and Polynomials

Denote by $\mathrm{\Gamma }$ the set of pointwise good sequences. Those are sequences of real numbers $(a_k)$ such that for any measure preserving flow $(U_t){t\in \mathbb R}$onaprobabilityspaceandforany$f\in L^\infty$,theaverages$\frac{1}{n} \sum{k=1}^{n} f(U_{a_k}x) $ converge almost everywhere. We prove the following two results. [1.] If $f:(0,\mathrm{\infty })\to \mathbb{R}$ is continuous and if $(f(ku+v){)}_{k\ge 1}\in \mathrm{\Gamma }$ for all $u,v>0$, then $f$ is a polynomial on some subinterval $J\subset (0,\mathrm{\infty })$ of positive length. [2.] If $f:[0,\mathrm{\infty })\to \mathbb{R}$ is real analytic and if $(f(ku){)}_{k\ge 1}\in \mathrm{\Gamma }$ for all $u>0$, then $f$ is a polynomial on the whole domain $[0,\mathrm{\infty })$. These results can be viewed as converses of Bourgain’s polynomial ergodic theorem which claims that every polynomial sequence lies in $\mathrm{\Gamma }$.