arXiv:2607.25454v1

A Criterion for Equidistribution along the $Ω$ Function over Polynomial Sequences with Applications

Zhi Qi, Cheng Zheng

math.DSmath.NT37A44

Paper metadata

Original submission: July 28, 2026
Claimed MSC: 37A44
arXiv categories: math.DS, math.NT
Paper license: arXiv non-exclusive distribution license ↗
Comments: 10 pages

Abstract

Let $P (Y_1, ..., Y_d)$ be a certain fixed homogeneous polynomial of integral coefficients. In this paper, we establish a quantitative equidistribution criterion for the ergodic averages along $Ω(|P (n_1, ..., n_d)|)$ . Consequently, by an estimate of Lachand, we prove the following variant of a theorem of Bergelson and Richter: if $P$ is an irreducible binary cubic form and $(X, T)$ is a uniquely ergodic system with unique invariant measure $μ$ , then for any $x \in X$ and $f \in C(X)$ , $\lim_{N \rightarrow \infty} \frac 1 {N^2} {\mathop{\sum\sum}_{n_1, n_2 \leqslant N}} f \big( T^{ Ω(|P (n_1, n_2)| ) } x \big) = \int_{X} f \ \mathrm{d} μ.$ Moreover, we prove in the appendix a related conjecture of Céspedes and Donoso over number fields.

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Audited against arXiv v1

Not a correctness certificate. A “Correct” result may include yellow typos or minor formal corrections that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.

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Detailed mathematical audit

Generated August 18, 2026

01Statements3 reported findingsCorrect

The equidistribution criterion and its cubic-form and number-field applications are correct. A local $p$ -adic lemma is misstated, but the conclusion needed downstream follows from a uniform derivative-valuation bound on the compact primitive $p$ -adic zero set.

Theorem ACorrect

General equidistribution criterion along $\Omega(|P(n)|)$

Pages 2–4 and 4–6 · Theorem A and Theorem 1 · arXiv:2607.25454v1

Under Hypotheses A and B with $\alpha>1/3$ and $\beta>1/2$ , Theorem A proves that the $\Omega(|P(n)|)$ -iterates are uniformly distributed in every uniquely ergodic system. Theorem 1 gives quantitative decay of the total variation between the distribution of $\Omega$ and its unit shift. Unique ergodicity converts that shift-invariance into the displayed orbit average. The removal of the primitive restriction is handled by Möbius inversion with an error controlled by Hypothesis B, so the normalized all-lattice-point average has the announced limit.

Theorem BCorrect

Irreducible cubic binary forms

Pages 2–3 · Theorem B · arXiv:2607.25454v1

Lachand's prime-value estimate supplies Hypothesis A for every irreducible integral cubic binary form with exponents on the required side of $1/3$ and $1/2$ . Such a form is factor-wise regular, so the repaired local-root argument supplies Hypothesis B. Substitution into Theorem A gives the $N^{-2}$ square average stated in Theorem B for every point and continuous observable of a uniquely ergodic system.

Theorem CCorrect

Imaginary-quadratic norm-form averages

Pages 3 and 12–16 · Theorem C and Appendix A · arXiv:2607.25454v1

For the listed class-number-one imaginary quadratic fields, Appendix A applies the Selberg–Delange expansion for the Dedekind zeta function to the norm-form counting region. The resulting shift-invariance of $\Omega$ is normalized by $C_P(N)$ , not by a square box. Boundary lattice points are lower order. Combining this arithmetic estimate with unique ergodicity gives the stated limit for every continuous $f$ and every starting point. The restriction to the fields listed in Definition 2 is retained in the theorem.

02Proofs3 reported findingsContains incorrect or incomplete proofs

Lemma 4 defines a smooth lifting level that need not exist when the reduction modulo $p$ is singular. The needed uniform lifting estimate is nevertheless recoverable by a different compactness-and-Hensel argument.

Section 2Correct and complete

Shift-invariance criterion and dynamical deduction

Pages 4–6 · Theorem 1 and proof of Theorem A · arXiv:2607.25454v1

The characteristic function of $\Omega(|P(n)|)$ is written as an Euler-product/sieve expression. Hypothesis A controls the prime contribution and Hypothesis B bounds values with excessive local multiplicity, producing the stated $L^1$ distance between $p_N(k+1)$ and $p_N(k)$ . Summation by parts against $f(T^k x)$ and the uniform ergodic theorem then show that every subsequential orbit distribution is $T$ -invariant; unique ergodicity identifies it with $\mu$ . The primitive-to-unrestricted passage uses an absolutely summable gcd decomposition.

Lemma 4Incorrect as written

A singular reduction does not become scheme-smooth at a higher congruence level

Pages 6–7 · Lemma 4 and its proof · arXiv:2607.25454v1

The proof defines $c_p$ as the first exponent at which a primitive zero becomes smooth modulo a higher power of $p$ . If the gradient vanishes modulo $p$ , increasing the modulus does not make that same reduction smooth, so the asserted definition can fail. Verified repair: over the compact set of primitive $p$ -adic zeros, smoothness over the $p$ -adic field implies that the minimum valuation among the partial derivatives is finite and uniformly bounded. Generalized Hensel lifting with that bound gives the uniform local estimate used later. This repair changes the lemma's argument but preserves all downstream theorems.

Lemma 2Correct after repair

Summation of local root counts after the Hensel repair

Pages 6–8 · proof of Lemma 2 · arXiv:2607.25454v1

For all but finitely many primes, projective smoothness gives the standard $p$ -adic lifting count. At each exceptional prime, the compactness bound on derivative valuations supplies a fixed loss independent of the lifting exponent. Multiplicativity over the irreducible factors and Chinese remaindering then bound the primitive root density by a divisor-type factor. Summing those local estimates over prime powers gives Hypothesis B with the strength needed in Section 2; no uniform smooth lifting level is required.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2607.25454v1
Authors listed: Zhi Qi, Cheng Zheng
Audit date: August 18, 2026

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A Criterion for Equidistribution along the ΩΩΩ Function over Polynomial Sequences with Applications

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

General equidistribution criterion along Ω(∣P(n)∣)\Omega(|P(n)|)Ω(∣P(n)∣)

Irreducible cubic binary forms

Imaginary-quadratic norm-form averages

Shift-invariance criterion and dynamical deduction

A singular reduction does not become scheme-smooth at a higher congruence level

Summation of local root counts after the Hensel repair

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A Criterion for Equidistribution along the $Ω$ Function over Polynomial Sequences with Applications

General equidistribution criterion along $\Omega(|P(n)|)$