This paper considers ergodic averages \(\frac 1t\sum_{n\leq t}f\circ T^{a_n}\), where \(a_n\) is \(a(n)\) or \(\lfloor a(n)\rfloor\) for a real-valued function \(a(x)\). It first defines a sequence \((a(n))\) to be universally good for norm convergence of \(L^p\) functions (i.e., norm good) if for each probability measure-preserving system \((\Omega,{\mathcal B},\mu,T)\) and \(f\in L^p(\mu)\), the above average converges in \(L^p\). Similarly, it defines \((a(n))\) to be universally good for pointwise convergence of \(L^p\) functions (i.e., pointwise good) if the above average converges for \(\mu\)-a.e. \(w\in\Omega\). The main results include a theorem which characterizes when a certain class of functions has \((a(n))\) and \((\lfloor a(n)\rfloor)\) norm good. This class of functions includes logarithmico-exponential, subpolynomial functions for which \(\lim_{x\to \infty}(x/a(x))=0\), and the criteria has to do with how close or how far the function is from a polynomial function. The pointwise good situation is more tricky; a theorem giving sufficient conditions and another one giving necessary conditions are given. This leaves open the question of an exact characterization of such functions, as is pointed out in Section 10, along with some other open questions. The final section provides some nice references of this material.