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