  The 0-1 knapsack problem is an important NP-hard problem that admits fully
polynomial-time approximation schemes (FPTASs). Previously the fastest FPTAS by
Chan (2018) with approximation factor $1+\varepsilon$ runs in $\tilde O(n +
(1/\varepsilon)^{12/5})$ time, where $\tilde O$ hides polylogarithmic factors.
In this paper we present an improved algorithm in $\tilde
O(n+(1/\varepsilon)^{9/4})$ time, with only a $(1/\varepsilon)^{1/4}$ gap from
the quadratic conditional lower bound based on $(\min,+)$-convolution. Our
improvement comes from a multi-level extension of Chan's number-theoretic
construction, and a greedy lemma that reduces unnecessary computation spent on
cheap items.
