  Denote by $\Gamma$ the set of pointwise good sequences. Those are sequences
of real numbers $(a_k)$ such that for any measure preserving flow $(U_t)_{t\in
\mathbb R}$ 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 $\big(f(ku+v)\big)_{k\geq 1}\in\Gamma$ 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
$\big(f(ku)\big)_{k\geq 1}\in\Gamma$ 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 $\Gamma$.
