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On multiplicative recurrence along linear patterns

T0 review · 2 major / 3 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read For every completely multiplicative f, liminf |f(an+b)-f(cn+d)|=0 exactly when a=c and a divides bd.

desk verdict Important new characterization, but the proof has a genuine gap in Proposition 4.1(iii) that must be repaired before the main theorem is trusted. read the letter →

arxiv 2412.03504 v2 pith:Q4PCEO6X submitted 2024-12-04 math.NT math.DS

Dimitrios Charamaras , Andreas Mountakis , Konstantinos Tsinas This is my paper · ORCID
classification math.NTmath.DS MSC 11N3737A44
keywords multiplicativerecurrencecompletelyfunctionsQ-trickChineseRemainderTheoremshiftsbinarycorrelationspretentiousdistancestrongaperiodicityfinitelygeneratedsystems
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

$\mathcal{M}$ is the class of completely multiplicative functions from $\mathbb{N}$ to the unit circle. The paper's goal is a complete answer to when two linear forms are multiplicatively independent: it claims that $\liminf_{n\to\infty}|f(an+b)-f(cn+d)|=0$ for every $f\in\mathcal{M}$ holds exactly when $a=c$ and either $b=d$ or $a\mid bd$. This matters because the same divisibility condition is the natural candidate for the set $\{(an+b)/(cn+d)\}$ to be a set of recurrence for multiplicative actions, and the theorem settles a family of examples that earlier work left open. The main equivalence also strengthens the conclusion: whenever the condition holds, the good values of $n$ form a set of positive upper logarithmic density. A companion result gives measurable recurrence along these ratios for all finitely generated multiplicative systems, and the pair version $(an+1,an)$ holds for all pairs of functions.

What carries the argument

The engine is a two-scale version of the Q-trick. Instead of averaging along $Qn$, the proof averages binary correlations $f^\ell(a_0Q^2n+a_0r_Q+b_0)f^\ell(vQ^2n+vr_Q+d_0)$ over a multiplicative Følner family $\Phi_K$, with each shift $r_Q$ chosen by the Chinese Remainder Theorem so that the congruences (20)--(21) hold. Those congruences force the two linear forms to be divisible in exactly the right way, so the correlation factors and the remaining pair is covered by the two-point correlation theorem for multiplicative functions. Pretentious pieces are simplified by a concentration estimate; non-pretentious pieces are forced to be strongly aperiodic along a suitable subsequence; and Proposition 4.1 yields a dichotomy in which the only possible nonzero correlation comes from modified characters. For finitely generated systems the same dichotomy is fed through the spectral representation of the action, while strong aperiodicity is proved for every finitely generated non-pretentious function.

What would settle it

Exhibit one completely multiplicative $f\colon\mathbb{N}\to S^1$ and one admissible quadruple with $a=c$ and $a\mid bd$ for which $\liminf_{n\to\infty}|f(an+b)-f(cn+d)|>0$, or for which the set $\{n: |f(an+b)-f(cn+d)|<\varepsilon\}$ has zero upper logarithmic density; the case $a=6, b=3, d=2$ is a concrete place to test.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is a dichotomy: for $(a,b,c,d)=1$, the liminf condition holds for every completely multiplicative function if and only if $a=c$ and ($b=d$ or $a\mid bd$). When the condition holds, for every $\varepsilon>0$ the set $A(f,\varepsilon)=\{n: |f(an+b)-f(cn+d)|<\varepsilon\}$ has positive upper logarithmic density; for pretentious or finitely generated $f$ it has positive lower logarithmic density. The only conceivable obstructions to small gaps are modified characters, that is, multiplicative functions that agree off finitely many primes with a Dirichlet character times $n^{it}$. This characterization is optimal, it subsumes the previously known rigidity result for the pair $(n+1,n)$, and it supplies positive evidence that $\{(6n+3)/(6n+2)\}$ is a set of multiplicative recurrence.

Load-bearing premise

The whole argument depends on the Chinese-Remainder shifts $r_Q$ satisfying the two congruence systems (20)--(21) for essentially every $Q$ in the multiplicative Følner family; if those congruences failed on a positive proportion of $Q$, the factorization step and the dichotomy that reduces correlations to modified characters would collapse.

Editorial extensions

If this is right

  • For every $a,b,d$ with $(a,b,d)=1$ and $a\mid bd$, and every completely multiplicative $f$, the set of $n$ with $|f(an+b)-f(an+d)|<\varepsilon$ has positive upper logarithmic density; in particular the known result for $(n+1,n)$ extends to all such pairs.
  • The earlier inconclusive example $\{(6n+3)/(6n+2)\}$ now has its required condition satisfied, so it is supported as a set of multiplicative recurrence.
  • Under the same divisibility assumption, $\{(an+b)/(an+d)\}$ is a set of recurrence for every finitely generated multiplicative system, with a positive lower density of good $n$.
  • For pairs of functions, $\liminf_{n\to\infty}|f(an+1)-g(an)|=0$ for every $a,f,g$, but the construct with $(n+2,n)$ shows no such pair theorem holds with a general shift $k$.
  • The equivalence identifies the exact necessary condition for Conjecture 1; what remains open is whether the divisibility condition is sufficient for all multiplicative systems, not merely finitely generated ones.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial extension: the proof isolates simultaneous strong aperiodicity as the only obstruction to the full conjecture; a concrete next step is to try to construct one aperiodic function whose powers cannot be made strongly aperiodic on a common sequence of scales.
  • Editorial extension: the two-scale CRT shift construction might transfer to ratios of higher-degree polynomials if an analogous divisibility condition can be written down; checking a quadratic analogue would show how much of the method is genuinely about linearity.
  • Editorial extension: the $(n+2,n)$ pair counterexample is built from two finite-valued modified characters modulo $4$; quantifying how large the minimal gap can be as a function of $k$ would make the obstruction quantitative.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 3 minor

Summary. The paper characterizes, for a, c in N and b, d in Z with gcd(a,b,c,d)=1, the condition that liminf_{n->infty} |f(an+b)-f(cn+d)|=0 holds for every completely multiplicative f:N->S^1. Theorem 1.1 asserts that this is equivalent to positive upper logarithmic density of the sets A(f,epsilon) and to the explicit condition a=c and (b=d or a|bd). Necessity is proved in Section 3 using modified Dirichlet characters; sufficiency is proved in Sections 4-5 using logarithmic averages over multiplicative Følner sequences, CRT-constructed shifts r_Q, Tao's two-point Elliott theorem, and Halász-type estimates. The paper also proves recurrence for finitely generated multiplicative systems (Theorem 1.5), a pair version (Theorem 1.2), and gives a counterexample showing that (n+2,n) fails for pairs of functions.

Significance. If the proof is repaired, Theorem 1.1 is a substantial and complete result: it subsumes the Klurman-Mangerel theorem for (n+1,n), generalizes the partial results of Donoso-Le-Moreira-Sun, and gives strong evidence toward their Question 7.2. Theorem 1.5 is a nontrivial extension to finitely generated multiplicative systems, and the counterexample in Section 6 is a useful contribution. The proof strategy is original and the paper is careful in stating open questions and in indicating exactly where the method stops for general systems. However, one load-bearing step in the proof of Proposition 4.1(iii) is invalid as written, and this gap propagates to the sufficiency direction of Theorem 1.1.

major comments (2)
  1. [Section 4, proof of Proposition 4.1(iii), final paragraph after Eq. (36)] The step 'if there is p not dividing a1 so that f(p) != p^{it_f} chi_{f,1}(p)chi_{f,1}(p), or equivalently f(p) != p^{it_f}' is false. In the g=f case the multiplier controlling the Følner shift is f(p)p^{-it_f}chi_{f,1}(p)^2, so the condition forcing the averages in (36) to vanish is f(p) != p^{it_f}chi_{f,1}(p)^2, not f(p) != p^{it_f}. When chi_{f,1}(p)^2 = -1 (possible for a Dirichlet character of order 4 whose conductor divides a1^infty, e.g. with a1=17), f(p)=p^{it_f} is compatible with f(p) != p^{it_f}chi_{f,1}(p)^2. The preceding argument actually yields f(p)=p^{it_f}chi_{f,1}(p)^2 for p not dividing a1, and the 'furthermore' conclusion f(n)=n^{it_f} for all n does not follow. This assertion is used in Section 5.2.4 for the powers in B2 and, more critically, in Section 5.3 at Eq. (60) to rule out exceptional pretentious powers g^m. The gap is therefore load-bearing for Proposition 5.3 and for (3)=>(1) in Theorem 1.1. No counterexample to the proposition is given, and a character-sum argument may repair the step, but as written the proof is incomplete.
  2. [Section 5.3, Eq. (60)] The proof of (60) invokes Proposition 4.1(iii) and concludes that the only exceptional pretentious case is g^m(n)=n^{it} for all n. Because the 'furthermore' in Proposition 4.1(iii) is unsupported, the exclusion of the modified-character exception is not justified. Without (60), the contradiction with (59) does not follow, and Proposition 5.3 is not proved. This is not a cosmetic issue: the same unsupported clause is also used in the proof of Proposition 5.2, so the sufficiency direction of Theorem 1.1 needs a genuine repair, not a local clarification.
minor comments (3)
  1. [Section 3, proof of Proposition 3.3, p=2 case] The displayed congruence 'anb ≡ and (mod 2^u)' appears corrupted; the intended statement is clearly a n b ≡ a n d (mod 2^u), and this should be corrected for readability.
  2. [Section 5.1, Eq. (38)] The inequality h_epsilon(x) >= epsilon + Re(sum_{1<=|ell|<R} c_ell e(ell x)) - epsilon^2 >= epsilon^2 + Re(...) is stated for x in [0,1] without explicitly saying that epsilon is taken small enough; this should be made explicit.
  3. [Section 6, Case 3] The notation in the paragraph after Eq. (68) switches between the original g and the modified functions ef and eg; this is understandable from context but would benefit from a sentence fixing the notation before the displayed formulas.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main theorem is derived from Tao's theorem and self-contained estimates; the sole self-citation is independent published support.

full rationale

The central derivation chain of the paper is self-contained and does not reduce to its own inputs. The necessary direction (2)=>(3) of Theorem 1.1 is proved by explicitly constructing modified Dirichlet characters that violate the liminf condition when a≠c or a∤bd; this is a direct argument, not a renaming or a fitted prediction. The sufficiency direction (3)=>(1) is built on Proposition 4.1, whose proof uses Tao's two-point correlation theorem (Theorem A), the concentration estimate Lemma 2.13, and the CRT-based construction of the shifts r_Q; none of these inputs contains the conclusion being proved. The separate case f^k = n^{it} is handled in Proposition 5.3 by a contradiction argument that again invokes Proposition 4.1; even if the skeptical objection about Proposition 4.1(iii) is correct, that is a mathematical gap in an intermediate lemma, not circularity, because the lemma is not assumed as its own conclusion. The only self-citation is [8, Lemma 2.6] in Section 7, used to assert that the spectral measure of a finitely generated multiplicative system is supported on Mfg. This is a published, independent theorem from the first author's prior work; it is load-bearing for Theorem 1.5 but it is not a restatement of any result proved in this paper and it does not assume the target result. Under the review rule, such an external published lemma is real evidence and does not raise the circularity score. No equation in the paper is shown to be equivalent to its own input by construction, and no fitted parameter is later called a prediction.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters or invented entities. The paper uses standard theorems from the literature; the only noteworthy reliance is a cited lemma from the first author's earlier paper in the final theorem, which is independent published work.

assumptions (5)
  • standard math Tao's two-point logarithmic Elliott/Chowla theorem (Theorem A, [30, Theorem 1.3])
    Invoked in Proposition 4.1 and repeatedly in Sections 5-7 to control binary correlations of multiplicative functions.
  • standard math Halász's theorem and Granville-Soundararajan pretentious distance formalism
    Basis for aperiodicity conclusions and Lemma 2.2/Corollary 2.3 in Section 2.
  • standard math Vinogradov-Korobov zero-free region for Dirichlet L-functions
    Used in Appendix A to prove strong aperiodicity of finitely generated functions (Lemma A.2).
  • standard math Bochner-Herglotz spectral theorem and [8, Lemma 2.6] that finitely generated systems have spectral measure supported on finitely generated multiplicative functions
    Used in Section 7 to pass from recurrence of averages of f to intersection of preimages. [8] is by the first author; it is a published lemma, not circular with this paper.
  • domain assumption Complete multiplicativity and unit-circle values for functions in M
    The whole paper works with f:N->S1 completely multiplicative, the natural class for the recurrence necessary condition.

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Cite this review

Pith. "Pith review of On multiplicative recurrence along linear patterns." pith.science (2026). https://pith.science/paper/Q4PCEO6X

@misc{pith2026241203504,
  author       = {Pith},
  title        = {Pith review of: On multiplicative recurrence along linear patterns},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Q4PCEO6X}},
  note         = {Machine review of arXiv:2412.03504}
}
abstract

In a recent article, Donoso, Le, Moreira and Sun studied sets of recurrence for actions of the multiplicative semigroup $(\mathbb{N}, \times)$ and provided some sufficient conditions for sets of the form $S=\{(an+b)/(cn+d) \colon n \in \mathbb{N} \}$ to be sets of recurrence for such actions. A necessary condition for $S$ to be a set of multiplicative recurrence is that for every completely multiplicative function $f$ taking values on the unit circle, we have that $\liminf_{n \to \infty} |f(an+b)-f(cn+d)|=0.$ In this article, we fully characterize the integer quadruples $(a,b,c,d)$ which satisfy the latter property. Our result generalizes a result of Klurman and Mangerel concerning the pair $(n,n+1)$, as well as some results of Donoso, Le, Moreira and Sun. In addition, we prove that, under the same conditions on $(a,b,c,d)$, the set $S$ is a set of recurrence for finitely generated actions of $(\mathbb{N}, \times)$.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Partition regularity of homogeneous quadratics: Current trends and challenges

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    A survey of methods proving partition regularity for pairs of variables in homogeneous quadratic equations, with the remaining cases reduced to an open conjecture about vanishing correlations.

Reference graph

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