Discrete spectrum and equicontinuity along sequences

This is the authors' abstract. We don't add key points for this paper.

This paper is devoted to the characterizations of discrete spectrum via sequential forms of equicontinuity and its (logarithmic) mean variants. In particular, it is shown that a measure-preserving system has discrete spectrum if and only if it is mean or logarithmic-mean equicontinuous, if and only if for any infinite sequence of non-negative integers, there exists a subsequence such that the system is equicontinuous, mean equicontinuous, or logarithmic-mean equicontinuous along any of its sub-subsequences; a transitive topological dynamical system has topological discrete spectrum if and only if it is mean or logarithmic-mean equicontinuous along every subsequence of non-negative integers, if and only if it is mean or logarithmic-mean equicontinuous when distal. Among other examples, we build a topological dynamical system that is logarithmic-mean equicontinuous along some sequence but not mean equicontinuous along the same one, and a measure-preserving system with discrete spectrum yet not mean equicontinuous along some sequence.