Asymptotics for partitions over the Fibonacci numbers and related sequences, II

Abstract

In a recent paper of the same title, we considered nondistinct partitions of integers over the Fibonacci sequence and related linearly recurrent sequences. In this paper, continuing this story, we apply the same method, but consider distinct partitions. In particular, let $q_F(m)$ denote the number of ways of writing $m$ as the sum of distinct Fibonacci numbers. It is well-known that, while $q_F(m)$ behaves wildly, its partial sums behave in a more structured way. In particular, Kempton showed that the function $N^{-\log 2/\logφ}\sum_{m\leqslant N}q_F(m)$ is asymptotically logarithmically-periodic. In this paper, we improve on this result by providing an explicit error term. Additionally, we prove the analogous result for a class of more general linear recurrent sequences.