Closed-form expected stopping time for composite moduli

Determine whether the expected stopping time E(T) of the Bernard–Letac sampler for composite modulus m admits a formula in terms of the Rényi entropies at the prime-power orders p_j^{a_j} appearing in the factorization of m.

Background

For prime modulus p, the paper derives an exact infinite-product formula for E(T) involving the power sums of the source distribution at orders p, p2, p3, and so on. No corresponding formula is established for composite m.

The obstruction is that the stopping set H_m is the intersection of the prime-power stopping sets, coupling digit and carry structures in different bases. Composite examples also exhibit unreachable stopping states and varying observable group sizes, motivating the question of whether a Rényi-entropy formula can nevertheless be obtained, perhaps through Möbius-type inversion.

References

The question remains whether a formula for $E(T)$ exists in terms of the R {e}nyi entropies at the prime-power orders $p_j{a_j}$.

— Algorithms, Complexity, and Entropy of the Bernard-Letac Fair-Sampling Construction  (2608.20234 - Gravel, 20 Aug 2026) in Open Problem P2, Section 5; see also Remark “No closed form for composite m”

It is an open question whether a data-adaptive stopping boundary can be designed that, after estimating $\pi$ online, dynamically selects a subset of $H_m$ to reduce $E(T)$ while maintaining fairness.

— Algorithms, Complexity, and Entropy of the Bernard-Letac Fair-Sampling Construction  (2608.20234 - Gravel, 20 Aug 2026) in Open Problem P4, Section 5; also stated in the proof discussion for Algorithm 2