Some Remarks on The Distribution of Subsequences of Second Order Linear Recurrences

Abstract

A common theme in mathematics is to examine a phenomenon in a local vs. global sense. For example, a group may have a particular property if and only if every subgroup has the same property. In number theory a given equation may be solvable in the integers if and only if it is solvable mod n for each n. With sequences, it is often the case that a sequence has a particular property (e.g. converges) if and only if each subsequence has a similar property. In this article we wish to examine the relationship between the uniform distribution of a second order linear recurrence and the behavior of certain types of subsequences. We begin with a few well-known facts about uniformly-distributed second order recurrences [1, 3, 4, 5, 6, 8].