For any epsilon, a dictionary storing S uses gap(S)(1+O(epsilon)) + O(n log gap(S)) bits and answers membership in O(log(1/epsilon)/log log(1/epsilon)) time, with a matching static lower bound.
Monotone minimal perfect hashing: searching a sorted table with O (1) accesses
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Optimal Time-Space Tradeoff for Dynamic Difference-Encoded Dictionaries
For any epsilon, a dictionary storing S uses gap(S)(1+O(epsilon)) + O(n log gap(S)) bits and answers membership in O(log(1/epsilon)/log log(1/epsilon)) time, with a matching static lower bound.