In this talk, we consider pointwise convergence of the Schrödinger means along sequences $t_n$ that converge to zero. We discuss sufficient conditions for the convergence and explain the key observation, which is that bounds on the maximal function $\sup_{n} |e^{it_n\Delta} f| $ can be deduced from those on $\sup_{0\lt t\le 1} |e^{it\Delta} f|$ when $\{t_n\}$ is contained in the Lorentz space $\ell^{r,\infty}.$ We will discuss sharp counterexamples for the related maximal estimates.