Information-theoretic interpretation of the p-adic limit

Determine whether the p-adic limit L(x)=lim_{n→∞} c(p^n x) associated with multinomial coefficients admits an interpretation in terms of the entropy of a probability distribution associated with x.

Background

The paper studies the p-adic limit L(x) obtained from the sequence of multinomial coefficients c(pn x), and records a product identity involving limits indexed by matrices with prescribed margins. Although this limit has algebraic and combinatorial structure, its possible information-theoretic meaning is not established.

The unresolved issue is whether one can associate a distribution with x so that L(x), or a natural transform of it, is described through an entropy quantity.

References

It remains to be determined whether $L(x)$ can be interpreted in terms of the entropy of a distribution associated with $x$.

— Algorithms, Complexity, and Entropy of the Bernard-Letac Fair-Sampling Construction  (2608.20234 - Gravel, 20 Aug 2026) in Open Problem P5, Section 5; introduced after Theorem 10 of Bernard–Letac