  Dictionaries have been one of the central questions in data structures. A
dictionary data structure maintains a set of key-value pairs under insertions
and deletions such that given a query key, the data structure efficiently
returns its value. The state-of-the-art dictionaries [Bender, Farach-Colton,
Kuszmaul, Kuszmaul, Liu 2022] store $n$ key-value pairs with only $O(n
\log^{(k)} n)$ bits of redundancy, and support all operations in $O(k)$ time,
for $k \leq \log^* n$. It was recently shown to be optimal [Li, Liang, Yu, Zhou
2023b].
  In this paper, we study the regime where the redundant bits is $R=o(n)$, and
show that when $R$ is at least $n/\text{poly}\log n$, all operations can be
supported in $O(\log^* n + \log (n/R))$ time, matching the lower bound in this
regime [Li, Liang, Yu, Zhou 2023b]. We present two data structures based on
which range $R$ is in. The data structure for $R<n/\log^{0.1} n$ utilizes a
generalization of adapters studied in [Berger, Kuszmaul, Polak, Tidor, Wein
2022] and [Li, Liang, Yu, Zhou 2023a]. The data structure for $R \geq
n/\log^{0.1} n$ is based on recursively hashing into buckets with logarithmic
sizes.
