MULTIPLY MINIMAL POINTS FOR THE PRODUCT OF ITERATES

Wen Huang, Song Shao, Xiangdong Ye

correctmedium confidence

Category: Not specified
Journal tier: Strong Field
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The paper proves the exact strong form the model judged likely open: it constructs a minimal weakly mixing system (arising from a horocycle system) in which, for every x, the diagonal (x,x) is transitive under T×T^2 but not minimal. This is Theorem A and is established via Theorem 2.1 together with Theorems 2.13–2.14 using Ratner’s theorems and a commensurator computation; the statement “for all x” is explicit. Hence the model’s “likely open as of cutoff” assessment is incorrect.

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} strong field

\textbf{Justification:}

The paper gives a definitive and elegant construction answering a natural question in topological dynamics by showing that a minimal weakly mixing system can have no multiply minimal points, with the stronger uniform property that every diagonal is transitive under T×T\^2 but not minimal. The argument skillfully blends arithmetic properties of lattices with Ratner theory. The manuscript is well organized; minor clarifications on standard properties invoked would further aid readers.