Möbius random law and infinite rank-one maps

E. H. el Abdalaoui, Cesar E. Silva

incompletemedium confidence

Category: Not specified
Journal tier: Specialist/Solid
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The paper states Theorem 4.1 claiming Möbius disjointness for every symbolic rank-one map under the sole hypothesis ∑ 1/p_n^2 = ∞, but its proof immediately imposes extra assumptions (weak mixing and an arithmetic non-divisibility condition for infinitely many primes) that are not present in the theorem statement. The argument then relies on these added hypotheses to obtain spectral singularity of prime-power dilates via a generalized Riesz-product criterion before invoking the DKBSZ criterion for Möbius disjointness. This mismatch indicates missing assumptions in the paper’s main claim (Theorem 4.1) and does not establish the unconditional result “under ∑ 1/p_n^2 alone,” which is exactly what the model says is (likely) open as of the cutoff. See the theorem statement and the added assumptions presented together on the same page, as well as the DKBSZ pathway and the Riesz-product machinery they use.

Referee report (LaTeX)

\textbf{Recommendation:} major revisions

\textbf{Journal Tier:} specialist/solid

\textbf{Justification:}

The manuscript uses an appealing Riesz-product and DKBSZ route, and it contains interesting infinite-measure formulations. However, the flagship finite/topological result (Theorem 4.1) is stated without hypotheses that the proof then crucially employs (weak mixing and an arithmetic prime non-divisibility condition). This undermines the correctness of the main claim under the bare hypothesis ∑1/p\_n\^2 = ∞. The work could become publishable after explicitly stating and justifying all needed assumptions and carefully verifying the spectral singularity step and its transfer to the system’s maximal spectral type.