Establish bit-cost analogues of the logarithmic law

Investigate and establish precise bit-cost analogues of the logarithmic relation between static dictionary covering and adaptive covering under optimized pointer representations.

Background

The logarithmic law proved in the paper concerns uniform phrase count, not encoded bit length. The paper emphasizes that dictionary references and self-references can have different costs and that greedy phrase-count minimization need not minimize bit length. It therefore leaves unresolved the corresponding quantitative law for optimized pointer-based bit-cost models.

References

Several avenues remain open for future work: closing the gap on the constant factor in preset-history bounds, extending the sandwich theorems to multi-word dictionaries, and investigating the precise bit-cost analogues of the logarithmic law under optimized pointer representations.

— On the Adaptive Covering Complexity of Strings: Unifying Relative and Self-Referential Compression  (2610.26021 - Marteau, 19 Oct 2026) in Conclusion, Section 6