{"id":"2528b743-fc23-4d66-b1a4-122a22c45a46","arxiv_id":"2412.03504","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For gcd(a,b,c,d)=1, the liminf of |f(an+b)-f(cn+d)| is zero for every completely multiplicative unit-circle f if and only if a=c and either b=d or a divides bd.","lead":"This paper determines exactly when the linear forms an+b and cn+d must take nearly equal values under every completely multiplicative function, infinitely often. The condition, a=c and either b=d or a divides bd, generalizes the Klurman-Mangerel theorem for consecutive integers and answers a question from Donoso, Le, Moreira and Sun.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 4.1(iii)'s 'furthermore' relies on a false equivalence; since Proposition 5.3 uses it, the proof of Theorem 1.1 needs repair before acceptance.","rationale":"After reading the proof, the most load-bearing point is not the robustness of the CRT congruences per se (Claims 1 and 2 appear correct and the exceptional sets are handled with the Følner property), but the derivation of the 'furthermore' statement in Proposition 4.1(iii). The proof contains a non-sequitur: 'or equivalently f(p)≠p^{it_f}' is false for quartic characters. This is not a cosmetic issue: Proposition 5.3 and, through it, the sufficiency direction (3)=>(1) of Theorem 1.1 rely on the 'furthermore' to exclude all pretentious powers except n^{it}. If the equivalence is not justified, the reduction to modified characters in part (iii) does not yield the claimed rigidity. The rest of the paper is careful, and the necessary direction (2)=>(3) is solid. The finitely generated recurrence result and the counterexample in Section 6 are independent and likely correct. I recommend CONDITIONAL acceptance: the authors should repair the 'furthermore' argument (or replace it with a direct character-sum vanishing argument) before the proof is complete.","tokens_in":43783,"tokens_out":35211,"duration_ms":308429,"concrete_test":"Independently compute, in the proof of Proposition 4.1(iii) with g=f, the ratio of the summands in (36) under Q→Qp: it is f(p)p^{-it_f}χ_{f,1}(p)^2. Then test the quartic example: take a1=17, a2=1, b1=1, b2=1, let χ be a character mod 17 of order 4 and define f∈M by f(p)=χ(p)^2 for p≠17, f(17)=1. Verify whether the Q-average in (36) is E_{Q∈Φ_K}χ^2(W), which tends to 0, while the claimed equivalence would predict a non-vanishing correlation unless f(p)=1. If the average is indeed 0, the proposition may still be true but the proof requires a different argument; if it is nonzero, Proposition 4.1(iii) is false.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Proposition 4.1(iii) (Section 4, final paragraph of the proof, after Eq. (36)) states that if g=f and the correlations do not tend to zero, then f(n)=n^{it_f} for all n. In the proof, the authors assert that if there is p∤a1 with f(p)≠p^{it_f}χ_{f,1}(p)χ_{f,1}(p), `or equivalently f(p)≠p^{it_f}`, then the averages in (36) vanish. This equivalence is incorrect: the multiplier relating the summands in (36) for Q and Qp is f(p)p^{-it_f}χ_{f,1}(p)^2, which equals 1 whenever χ_{f,1}(p)^4=1, even when χ_{f,1}(p)^2≠1. Dirichlet characters of order 4 (possible when a1 has a prime ≡1 mod4, e.g., a1=17) satisfy this. The subsequent conclusion that χ_f is principal relies on this equivalence. This 'furthermore' is used in Proposition 5.3 to rule out exceptional pretentious powers in (60); without it, the proof of (3)=>(1) in Theorem 1.1 has a gap. No counterexample is known, and a character-sum argument may rescue the claim, but as written the step is invalid.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":44040,"tokens_out":17771,"duration_ms":162914,"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":[{"comment":"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.","section":"Section 4, proof of Proposition 4.1(iii), final paragraph after Eq. (36)"},{"comment":"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.","section":"Section 5.3, Eq. (60)"}],"minor_comments":[{"comment":"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.","section":"Section 3, proof of Proposition 3.3, p=2 case"},{"comment":"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.","section":"Section 5.1, Eq. (38)"},{"comment":"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.","section":"Section 6, Case 3"}],"recommendation":"major_revision","confidential_remarks":"To the editor: the flaw I identify is confined to a dense technical proposition, but it is genuinely load-bearing: it is used in the proofs of both Proposition 5.2 and Proposition 5.3. I see no reason to doubt the main theorem, and a revision that proves the needed 'furthermore' statement (or replaces it with a correct substitute) could well make the paper acceptable. I would therefore recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"New here is real: Theorem 1.1 gives the first complete characterization of the quadruples (a,b,c,d) for which liminf |f(an+b)-f(cn+d)| = 0 for all completely multiplicative f, settling the necessary-condition side of the authors' Question 1 and subsuming the Klurman–Mangerel (n,n+1) theorem and the Donoso–Le–Moreira–Sun sufficient conditions. The Q^2-trick with CRT-chosen shifts is a genuine new idea, and the counterexample for (n+2,n) sharpens the boundary for pairs of functions. The finitely generated recurrence result is a nice bonus.\n\nBut there is a load-bearing flaw in Proposition 4.1(iii). In the final paragraph, the paper asserts that for g=f, if some p∤a1 satisfies f(p) ≠ p^{it_f}χ_{f,1}(p)χ_{f,1}(p), 'or equivalently' f(p) ≠ p^{it_f}, then the averages in (36) vanish. That equivalence is false when χ_{f,1} has order 4 (possible if a1 has a prime ≡1 mod 4). The argument only yields f(p)=p^{it_f}χ_{f,1}(p)^2 for every p∤a1, not f(p)=p^{it_f}. The subsequent conclusion that χ_f is principal does not follow, and the 'furthermore' — that f(n)=n^{it} — is not established. Proposition 5.3 leans on exactly this strong conclusion to rule out exceptional pretentious powers in (60); the proof of (3)=>(1) therefore has a gap as written. I don't see a counterexample to the theorem, and a character-sum argument may well repair the step, but it is a real gap, not a typo.\n\nMinor remark: Remark 2 overstates the counterexample: it proves failure for k=2, not for every k.\n\nThe rest of the structure — the aperiodic/pretentious split, use of Tao's two-point theorem, the finitely generated case — looks coherent to me, though the congruence details in Section 4 are dense and I did not mechanically verify them all. This paper deserves a serious referee: the result is important and the flaw looks repairable, but the referee must push hard on Proposition 4.1(iii) and its application in Proposition 5.3. I would not cite it as a proven theorem until that step is fixed.","headline":"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.","tokens_in":44597,"tokens_out":6833,"would_cite":false,"duration_ms":59533,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11N37","37A44"],"pacs":[],"model":"deepseek-v4-flash","headline":"For every completely multiplicative f, liminf |f(an+b)-f(cn+d)|=0 exactly when a=c and a divides bd.","keywords":["multiplicative recurrence","completely multiplicative functions","Q-trick","Chinese Remainder Theorem shifts","binary correlations","pretentious distance","strong aperiodicity","finitely generated systems"],"falsifier":"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.","tokens_in":43580,"feed_emoji":"♾️","tokens_out":10990,"duration_ms":98379,"temperature":0.7,"pith_summary":"$\\mathcal{M}$ 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.","feed_headline":"Multiplicative gaps vanish exactly when a=c and a|bd","feed_subtitle":"For every unit-circle multiplicative f, liminf |f(an+b)-f(cn+d)|=0 precisely under this condition.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Proves the rigidity result for the pair (n+1,n) that Theorem 1.1 extends to all pairs with a=c and a|bd.","marker":"[22]"},{"why":"Studies multiplicative recurrence for sets {(an+b)/(cn+d)} and poses the question whose necessary condition this paper resolves.","marker":"[9]"},{"why":"Introduces the original Q-trick and the multiplicative Følner family used as the basis of the present two-scale argument.","marker":"[13]"},{"why":"Supplies the two-point correlation theorem for multiplicative functions used to kill non-pretentious pieces of the averages.","marker":"[30]"},{"why":"Provides the subsequence lemma used to make a single non-pretentious function behave strongly aperiodic on a chosen scale.","marker":"[10]"},{"why":"Shows the relevant correlation limits along progressions exist for pretentious functions, enabling lower logarithmic density conclusions.","marker":"[21]"},{"why":"Establishes that finitely generated multiplicative systems have spectral measures supported on finitely generated multiplicative functions, used in Theorem 1.5.","marker":"[8]"}],"fun_headline_variants":["Multiplicative gaps vanish iff a=c and (b=d or a|bd)","Zero liminf for all f iff a=c, with b=d or a|bd","a=c, b=d or a|bd: exact vanishing gap condition","Gaps vanish for every f iff a=c and (b=d or a|bd)"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Multiplicative gaps vanish iff a=c and (b=d or a|bd)","Zero liminf for all f iff a=c, with b=d or a|bd","a=c, b=d or a|bd: exact vanishing gap condition","Gaps vanish for every f iff a=c and (b=d or a|bd)"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001046,"raw_usage":{"total_tokens":4396,"prompt_tokens":945,"completion_tokens":3451,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":561,"completion_tokens_details":{"reasoning_tokens":3364}},"tokens_in":561,"tokens_out":3451,"duration_ms":25787,"temperature":1.0,"reasoning_tokens":3364,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T22:19:38.385385+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Klurman and A","cited_arxiv_id":null,"evidence_quote":"Proves the rigidity result for the pair (n+1,n) that Theorem 1.1 extends to all pairs with a=c and a|bd."},{"cited_title":"Donoso, A","cited_arxiv_id":null,"evidence_quote":"Studies multiplicative recurrence for sets {(an+b)/(cn+d)} and poses the question whose necessary condition this paper resolves."},{"cited_title":"Frantzikinakis, O","cited_arxiv_id":null,"evidence_quote":"Introduces the original Q-trick and the multiplicative Følner family used as the basis of the present two-scale argument."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the two-point correlation theorem for multiplicative functions used to kill non-pretentious pieces of the averages."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the subsequence lemma used to make a single non-pretentious function behave strongly aperiodic on a chosen scale."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Shows the relevant correlation limits along progressions exist for pretentious functions, enabling lower logarithmic density conclusions."},{"cited_title":"Charamaras","cited_arxiv_id":null,"evidence_quote":"Establishes that finitely generated multiplicative systems have spectral measures supported on finitely generated multiplicative functions, used in Theorem 1.5."}],"review_version":1}