arXiv:2610.15809v1

A note on the $p$ -adic inhomogeneous Littlewood conjecture

math.NT11J8311K41

Paper metadata

Original submission: October 11, 2026
Claimed MSC: 11J83, 11K41
arXiv categories: math.NT
Paper license: CC BY 4.0 ↗

Abstract

Bugeaud, Haynes and Velani (2011) resolved a metric version of the $p$ -adic Littlewood conjecture. We establish a fully inhomogeneous generalisation of their result, as well as a partially inhomogeneous generalisation of a result of Harrap and Haynes (2013) in the more general mixed setting.

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Audited against arXiv v1

Not a correctness certificate. A green “no issues found” result may include yellow suspected typos or minor formal issues 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.

Current report

Detailed mathematical audit

Generated October 24, 2026

01Statements2 reported findingsNo statement issues found

The fully inhomogeneous fixed-prime theorem and the mixed-sequence extension follow from valid lower bounds for the totient-weighted divergence sums and the cited weak inhomogeneous Duffin–Schaeffer theorem. Two finite-domain conventions need local textual corrections, but neither changes a conclusion.

Theorem 1.9No issue found

The shifted prime-adic factors supply the required logarithmic multiplicity

PDF pp. 2–4 · arXiv:2610.15809v1

For each vector of valuations with $p_j^{\ell_j}\le N^{1/(2k)}$ , the Chinese remainder parametrization leaves a fixed positive proportion of an interval of length $N/(p_1^{\ell_1}\cdots p_k^{\ell_k})$ . Lemma 2.1 gives the same-scale lower bound for the average of $\varphi(n)/n$ . Multiplication by the reciprocal prime-adic factors cancels the denominator product term by term; the $\asymp(\log N)^k$ valuation vectors then give the divergence criterion required by the cited theorem.

Theorem 3.3No issue found

The mixed-sequence argument has the stated divergence strength

PDF pp. 5–6 · arXiv:2610.15809v1

On the layer $d_\ell\mid n$ and $d_{\ell+1}\nmid n$ , supermultiplicativity of the totient and the AM–GM bound give $\sum\varphi(n)/n\gg\varphi(d_\ell)N/d_\ell^2$ . Summing through half the available divisibility chain yields $N\psi(2N)L$ . Since strict divisibility makes $d_{\ell+1}\ge2d_\ell$ , the counting function satisfies $L(2N)\le L(N)+1$ , which justifies the condensed-series step even though its product with $\psi$ need not itself be monotone.

02Proofs4 reported findingsNo proof issues found

The totient averaging and dyadic summations are correct, and the external metric theorem has exactly the fixed-shift, unrestricted-numerator, finite-valued scope required. The proof has two harmless domain conventions and one displayed binding typo that should be corrected locally.

Definition of

\Psi

Suspected minor formal issue

Positive shifts can make the displayed auxiliary function undefined at finitely many integers

PDF p. 3, opening of Section 2 · arXiv:2610.15809v1

The proof sets $\Psi(n)=\psi(n)/(\lvert n-n_1\rvert_{p_1}\cdots\lvert n-n_k\rvert_{p_k})$ . If some $n_j$ is a positive integer and $n=n_j$ , one factor is zero, whereas Theorem 1.11 requires a finite-valued function. Define $\Psi(n)=0$ on the finite set $\{n_1,\ldots,n_k\}\cap\mathbb N$ and use the displayed formula elsewhere. Removing finitely many denominators affects neither divergence nor the infinitely-many-solutions conclusion.

Final display in Section 2Suspected typo

A valuation-dependent product is printed outside its binding sum

PDF p. 4, final calculation · arXiv:2610.15809v1

The factor $p_1^{\ell_1}\cdots p_k^{\ell_k}$ is printed before $\sum_{\ell\in W(N)}$ , although it depends on the bound variable $\ell$ , and the next line drops the sum. Keep that factor inside the $\ell$ -sum and apply Lemma 2.1 term by term. Each term is $\gg N\psi(2N)$ , and $\lvert W(N)\rvert\asymp(\log N)^k$ , producing exactly the printed final bound.

Definition of

\lvert\cdot\rvert_D

Suspected minor formal issue

The pseudo-absolute value needs the usual initial divisor

PDF p. 5, Section 3.2 · arXiv:2610.15809v1

For the stated class of strictly increasing divisibility sequences, the set $\{d_\ell^{-1}:d_\ell\mid n\}$ can be empty when $d_1>1$ . Prepend $d_0=1$ and take the minimum over $\ell\ge0$ , equivalently impose the standard convention that the chain begins with $1$ . This makes the definition total on $\mathbb N$ and leaves every large-index layer and every estimate in Theorem 3.3 unchanged.

Lemma 2.1 and proof of Theorem 3.3No issue found in the reviewed proof

Both AM–GM estimates have uniform positive lower bounds

PDF pp. 3–6 · arXiv:2610.15809v1

In Lemma 2.1 the excluded residue primes form the fixed product $q=p_1\cdots p_k$ , while for every other prime the normalized divisibility frequency is at most $p^{-1}+N^{-1/2}\ll p^{-1/2}$ ; hence the logarithm of the geometric mean is bounded by $\sum_p p^{-3/2}$ . The mixed proof similarly has frequency at most $2/p$ , leading to $\sum_p p^{-2}$ . These bounds are uniform in the valuation vector or layer to which they are applied.

03Novelty0 reported findingsNo novelty concerns found

No non-novelty findings.

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Paper: arXiv:2610.15809v1
Authors listed: Sam Chow, Owen Jones
Audit date: October 24, 2026

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A note on the ppp-adic inhomogeneous Littlewood conjecture

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

The shifted prime-adic factors supply the required logarithmic multiplicity

The mixed-sequence argument has the stated divergence strength

Positive shifts can make the displayed auxiliary function undefined at finitely many integers

A valuation-dependent product is printed outside its binding sum

The pseudo-absolute value needs the usual initial divisor

Both AM–GM estimates have uniform positive lower bounds

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A note on the $p$ -adic inhomogeneous Littlewood conjecture