Moroz’s Engel–Pierce expansion conjectures

Let $G$ denote Moroz's map from modified Engel expansions to Pierce expansions, associated with the digit correspondence $p_n=e_n+n-2$ , where $(e_n)$ is the Engel digit sequence and $(p_n)$ is the corresponding Pierce digit sequence. Let $\mathcal{E}$ be the exceptional set specified in Moroz's formulation. The conjectures assert that $G$ is continuous at every point of its domain outside $\mathcal{E}$ , equivalently $\operatorname{Disc}(G)\subseteq\mathcal{E}$ , and that $G$ is nowhere monotone: for every nondegenerate interval $I$ in its domain, the restriction $G\!\restriction_I$ is not monotone.

References

Primary source

arXiv

Additional references

Baire-category transfer between Engel and Pierce expansions — arXiv — Min Woong Ahn

Posted 5 days ago · edited 5 days ago

Progress summary

Refreshed 5 days ago

Claimed solved

A new preprint claims to prove both of Moroz’s conjectures about how Engel and Pierce expansions correspond, but the claim has not been independently verified.

Moroz’s conjectures concern the correspondence between Engel and Pierce expansions, including its continuity behavior and an associated exceptional set. The latest preprint claims an explicit construction that establishes both conjectured regularity properties.

October 2026 claimed resolution

Min Woong Ahn’s preprint constructs the correspondence, describes its discontinuities, and claims proofs of both conjectures. This is a claimed resolution, not an independently verified result; the retrieved related literature does not establish these specific statements.

Current status (as of October 2026): Both conjectures are claimed solved by Ahn’s preprint, but independent verification of the specific construction and proofs is not recorded.

Sources

No solutions have been posted yet.