{"id":"89f8a154-ef47-48bf-9860-c9548486ddc2","arxiv_id":"1908.02732","paper_version":2,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Correlations of multiplicative functions along deterministic low-complexity sequences and along independent sequences vanish under aperiodicity and independence assumptions, proved via ergodic-theoretic disjointness and characteristic factors.","lead":"A mathematician proves new correlation results for multiplicative functions like the Liouville function along deterministic and independent sequences, using ergodic theory to turn analytic number theory input into structural statements about measure-preserving systems. The results extend recent theorems of Tao and Teravainen and the author's own prior work, and include cases such as correlations of Liouville along sequences like floor(n alpha + beta).","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.1 collapses to Theorem 3.3, a broad disjointness statement delegated to [7,8] and not reproved; if that statement is false or narrower than stated, the deterministic-sequence results fail.","rationale":"The paper's own machinery is clear, and the reductions from Theorems 1.1, 1.3, and 1.5 to ergodic statements are well organized. I checked the proof of Proposition 3.2: the use of logarithmic averages, the continuity of F on X×(Y\\Z*), and the identification of the joining as a product are coherent. The main theorems are not self-contained; Theorem 3.3 and Theorem 4.3 are imported. Of these, Theorem 3.3 carries the full weight of the deterministic-sequence results (Theorem 1.1, Corollary 1.2, Theorem 1.6). Theorem 4.3 underpins Theorems 1.3 and 1.5, but the reader's verdict already centers on Theorem 3.3, and I agree. The obstacle in Section 1.6 does not refute Theorem 3.3, but it shows why one cannot naively extend the argument to f∘a, which underscores that the proof relies on the exact structural statement for uncomposed functions. Since [7] and [8] are published in Annals and IMRN, the risk is probably low, but this paper deliberately does not reprove them and the central claim is conditional on them. Hence CONDITIONAL is the right verdict; no adjustment is needed.","tokens_in":23810,"tokens_out":9908,"duration_ms":115895,"concrete_test":"Obtain the published statements of [7, Proposition 3.12] and [8, Theorem 1.5] and reconstruct the proof of Theorem 3.3 line by line, with special attention to two points: (i) the class of multiplicative functions allowed (unit-disc versus real-valued, and whether any aperiodicity or mean condition is imposed), and (ii) whether the zero-entropy system must be totally ergodic or merely ergodic, and where that hypothesis enters. If both published statements hold verbatim in the needed generality, Theorem 3.3 is established and the concern is settled; if either imposes an extra hypothesis not satisfied by arbitrary U-valued multiplicative functions, Theorem 1.1 needs qualification or a new argument.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 1.1 is a direct reduction: Theorem 3.1 (proved as Proposition 3.2) converts the composed correlation into an average of uncomposed correlations, and Theorem 3.3 is the only step that transfers vanishing from identity shifts to deterministic totally ergodic shifts. Theorem 3.3 states that every F-system of arbitrary U-valued multiplicative functions is disjoint from every zero-entropy totally ergodic system, and it is quoted from [8, Theorem 1.5] and [7, Proposition 3.12] with no proof. The paper's Section 1.6 flags a nearby obstruction: the analogous spectrum statement for f o a cannot be proved because an F-system of lambda might be the T(x,y)=(x,y+x) system; that does not disprove Theorem 3.3 but shows the boundary is delicate. The load-bearing question is exactly the generality of the cited disjointness result: does it apply to all bounded multiplicative functions with values in the unit disc, or only to a subclass, such as real-valued, strongly aperiodic, or nonzero-mean functions? Theorem 1.1 and Corollary 1.2 need the full U-valued statement for products f1...fℓ and for arbitrary Dirichlet characters; a single missing hypothesis in [7, Proposition 3.12] would leave the central claim unproven. This is a verification gap rather than a discovered error; the internal argument from Theorem 3.3 onward is coherent and the reductions in Sections 3 and 4 appear sound.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies logarithmic correlations of multiplicative functions along deterministic and independent sequences. The main results are: Theorem 1.1 extends the logarithmically averaged Chowla/Elliott-type vanishing results of Tao and Tao-Teräväinen from the identity sequence to compositions with deterministic totally ergodic sequences; Theorems 1.3 and 1.4 establish vanishing of correlations with shifts in sets or sequences that satisfy linear independence conditions; Theorem 1.5 establishes a product formula for real-valued multiplicative functions with weakly independent, jointly equidistributed shifts; and Theorems 1.6–1.8 deduce sign-pattern results. The deterministic results are proved via a reduction (Theorem 3.1) to a disjointness statement (Theorem 3.3) between F-systems of multiplicative functions and zero-entropy totally ergodic systems, with Theorem 3.3 quoted from the author's prior work with Host. The independent-sequence results are proved via ergodic-theoretic statements (Propositions 4.4 and 4.5) using characteristic factors and equidistribution on nilmanifolds.","tokens_in":24067,"tokens_out":25698,"duration_ms":265672,"significance":"Assuming the quoted structural results are correct at the stated generality, the paper contains substantial and interesting new results: it transfers logarithmically averaged Chowla-type vanishing to deterministic totally ergodic shifts and proves general theorems for independent shifts, including sign-pattern results. The new ergodic statements are proved in detail and the reduction structure is elegant. The paper also honestly documents the limitation that analogous statements for f∘a are unknown and face a concrete obstacle (Section 1.6). However, the deterministic part is fully conditional on the imported disjointness theorem, and the independent-sequence part contains a false intermediate theorem (Theorem 4.1). These issues must be resolved before the claims can be accepted.","major_comments":[{"comment":"Theorem 4.1 is false as stated. In the proof of Theorem 4.1 assuming Theorem 1.3, it is claimed that lim_{n→∞, n∈R} |(a_1(n),...,a_ℓ(n))| = ∞ because otherwise a fixed value would occur infinitely often and would contradict the assumption that S = {(a_1(n),...,a_ℓ(n)): n∈R} has independent elements. This is incorrect: the definition of 'independent elements' concerns solutions in S, and if a fixed vector b ∈ N^ℓ occurs for infinitely many n ∈ R, then S simply contains b once, so the condition is not violated. For example, take ℓ = 2, r = 1, R = N, a_1(n) = a_2(n) = 1, f_0 = 1, f_1 = f_2 = λ, and any sequence of intervals M. Then S = {(1,1)} has independent elements (the equation k·n = 0 has at most one solution in S), but the expression in (24) is Elog_{m∈M} λ(m+1)^2 = 1 for every n, so the asserted limit is 1, not 0. The statement of Theorem 4.1 needs a strengthened hypothesis (e.g., that the a_j are independent sequences, or that |(a_1(n),...,a_ℓ(n))| → ∞ along R), and the proof of Theorem 1.4 must be adjusted accordingly.","section":"Section 4, Theorem 4.1 and its proof"},{"comment":"Theorem 3.3 is the load-bearing step in the proof of Theorem 1.1, but it is quoted without proof and without stating the precise results from [8, Theorem 1.5] and [7, Proposition 3.12] that imply it. Since Theorem 1.1 requires the disjointness statement for joint F-systems of arbitrary collections of U-valued multiplicative functions, the authors should state the imported theorems explicitly and explain how they combine, or provide a proof. In particular, it should be confirmed that [7, Proposition 3.12] applies to joint F-systems of collections of U-valued multiplicative functions, not only to real-valued or single-function systems. This is a verification gap in the current manuscript.","section":"Section 3, Theorem 3.3"}],"minor_comments":[{"comment":"The abstract contains a typographical line break: 'taken al ong de-terministic' should read 'taken along deterministic'.","section":"Abstract"},{"comment":"In the statement of Theorem 4.7, the phrase 'the following limit exists in L2(μ)' is followed by an expression that ends with 'dμ', which is misplaced; the limit should be of the average Ep∈Pd ∏_{j=1}^ℓ T^{pj}F_j (without the trailing 'dμ'), or the integral should be written properly.","section":"Section 4, Theorem 4.7"},{"comment":"The remark defining deterministic sequences for finite-valued sequences using Cesàro averages is somewhat convoluted and could benefit from a clearer statement of the word-complexity condition.","section":"Section 1.2, definition of deterministic sequence"},{"comment":"In the proof of Proposition 4.4 for rotations, the assertion that the sequence (pβ_n)_{p∈P_d} is equidistributed on T for irrational β_n is used without a reference; adding a citation for equidistribution of primes in arithmetic progressions would be helpful.","section":"Section 4, proof of Lemma 4.10"},{"comment":"In the proof of Theorems 1.3 and 1.5 assuming Propositions 4.4 and 4.5, the argument relies on the fact that strong aperiodicity implies orthogonality to the Kronecker factor via Proposition 4.2(i); it would be clearer to state explicitly that this gives orthogonality to the rational Kronecker factor needed in Proposition 4.4.","section":"Section 4.2, proof of Theorem 1.5"}],"recommendation":"major_revision","confidential_remarks":"The paper is a strong contribution conditional on the imported structural results, but the false statement of Theorem 4.1 must be corrected. The heavy reliance on the author's own prior work with Host is a concern for evaluation of the novelty, but the transfer results are genuinely new. I recommend a thorough revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a genuine extension of recent logarithmic Chowla/Elliott results, and the main open question after reading is not whether the new proofs are right but how much weight one is willing to put on a quoted structural theorem.\n\nWhat is new: Theorem 1.1 transfers Tao's two-point and Tao–Teräväinen's odd-order vanishing from a(n)=n to any deterministic totally ergodic sequence a, provided the product f1...fℓ is non-pretentious to every Dirichlet character. That is a clean, significant step beyond the identity sequence. The independent-sequence results in Theorems 1.3–1.5 cover shifts in independent sets and weakly independent equidistributed sequences, including a genuinely new identity for real-valued multiplicative functions in Theorem 1.5 that fails for complex-valued ones. The reductions in Section 4 — the characteristic factor arguments and the induction on nilmanifold step — are carried out in the paper, not just quoted, and they look sound to me. Corollary 1.2 gives unbounded partial sums for λ∘a when a is deterministic and totally ergodic, which is a nice dividend.\n\nThe soft spot is the one the stress test and reader's report both identify: the deterministic case rests entirely on Theorem 3.3, the blanket disjointness of all F-systems of bounded multiplicative functions from zero-entropy totally ergodic systems, quoted from [7,8] with no proof here. That is a normal division of labor — those theorems are published, one in Annals — but it does mean Theorem 1.1 is only as good as that theorem's full U-valued generality. I found no concrete reason to doubt it, and the paper's Section 1.6 is unusually candid about the nearby obstacle for composed F-systems. A referee should check that [7, Proposition 3.12] and [8, Theorem 1.5] really cover the exact class of functions used here, including products and Dirichlet characters. That is a verification task, not a discovered error.\n\nThe citation pattern is heavy on the author's own prior work, but for once that is warranted: the F-system machinery is genuinely his and Host's, and the competing Tao/Teräväinen results are cited and used fairly. Minor point: the sign-pattern corollaries are routine derivatives of earlier arguments, not the core novelty.\n\nBottom line: this paper deserves a serious referee. The reductions are coherent, the new parts are real, and the main architectural risk is openly delegated rather than hidden. Send it out.","headline":"Extends logarithmic Chowla/Elliott zero results to deterministic totally ergodic compositions, with the heavy lifting delegated to the author's earlier disjointness theorems — a real contribution, honestly limited.","tokens_in":24649,"tokens_out":2399,"would_cite":true,"duration_ms":28115,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11N37","37A45","11K65"],"pacs":[],"model":"deepseek-v4-flash","headline":"Correlations of multiplicative functions along deterministic sequences vanish, extending known vanishing results through a transfer identity powered by ergodic disjointness.","keywords":["multiplicative functions","Liouville function","Chowla conjecture","Elliott conjecture","deterministic sequences","ergodic theory","logarithmic averages","Furstenberg correspondence"],"falsifier":"Compute the logarithmic average of $\\lambda([m\\alpha])\\lambda([(m+1)\\alpha])$ for an irrational $\\alpha>1$; the theorem predicts the limit is $0$, so a persistent non-zero limit would disprove Theorem 1.1. A more conceptual test: exhibit any bounded multiplicative function whose associated measure-preserving system has non-trivial irrational spectrum (the paper's own candidate is the system $(x,y)\\mapsto(x,y+x)$ on $\\mathbb{T}^2$), which would break the quoted disjointness theorem and, for $a(n)=[n\\alpha]$, put $e^{2\\pi i\\alpha}$ in the spectrum of the composed sequence.","tokens_in":23555,"feed_emoji":"🎲","tokens_out":17204,"duration_ms":162999,"temperature":0.7,"pith_summary":"The paper establishes that the conjectured random behaviour of multiplicative functions—above all the Liouville function—persists when the arguments are run through a broad class of deterministic sequences, and that correlations across linearly independent shifts vanish or factor as expected. The headline theorem (Theorem 1.1) says that if the deterministic sequence is totally ergodic (for instance, $a(n)=[n\\alpha+\\beta]$ with irrational $\\alpha>1$) and the product of the multiplicative functions is not weakly pretentious to any Dirichlet character (meaning its logarithmic average distance to every periodic completely multiplicative character remains positive), then the logarithmically averaged correlation $\\prod_{j=1}^{\\ell} f_j(a(m+n_j))$ converges to $0$. The proof is built on a correlation identity (Theorem 3.1) that expresses the composed correlation as a double average of ordinary correlations; this identity follows from disjointness of the associated measure-preserving systems of multiplicative functions from zero-entropy totally ergodic systems. A parallel set of results (Theorems 1.3, 1.4, 1.5) treats shifts taken from independent or equidistributed families, with the same vanishing or factorization conclusions, using characteristic factors, nilmanifold equidistribution, and short-interval results. Taken together these are the strongest currently known partial realizations of the conjectured vanishing and factorization laws for multiplicative functions along low-complexity sequences and thin shift sets.","feed_headline":"Liouville correlations vanish along deterministic sequences","feed_subtitle":"Transfer identity pushes known vanishing results to shifts like [nα], and yields sign-pattern equidistribution.","key_machinery":"The load-bearing object is the F-system of a bounded sequence: the shift-invariant measure on the space of sequences obtained as a weak-star limit of logarithmic orbit averages. Theorem 3.1 is the key identity: it equates the logarithmic correlation of the composed sequences $f_j\\circ a$ with an iterated logarithmic correlation of the original $f_j$, and it is derived from disjointness (Theorem 3.3), which forces the relevant joining to be a product measure. A second mechanism, Theorem 4.3, expresses ordinary logarithmically averaged correlations of multiplicative functions as an average over primes in a residue class of integrals of shifted products; this is what reduces the independent-shift theorems to purely ergodic statements (Propositions 4.4 and 4.5), which are proved by induction on the step of a nilsystem using nilcharacters and an equidistribution lemma for multi-parametric nilorbits (Lemma 4.12).","core_discovery":"The central discovery is a correlation identity: for a deterministic totally ergodic $a$ and multiplicative functions $f_1,\\dots,f_\\ell$ with values in the unit disc, after passing to a subsequence of intervals $M'$ we have $$$E^{{\\log}}$_{m\\in M'}\\prod_{j=1}^{\\ell} f_j(a(m+n_j)) = $E^{{\\log}}$_{n\\in M'}$E^{{\\log}}$_{m\\in M'}\\prod_{j=1}^{\\ell} f_j(m+a(n+n_j)).$$ Once this holds, the known vanishing theorems for ordinary shifts apply directly, because for each fixed $n$ the arguments $a(n+n_j)$ are just fixed points. The identity is powered by Theorem 3.3: every measure-preserving system attached to a bounded collection of multiplicative functions (its F-system) is disjoint from every zero-entropy totally ergodic system, so the joining used to relate the two averages is necessarily the product joining. For independent shifts, the paper proves that correlations with at least one strongly aperiodic factor vanish as $|n|\\to\\infty$ along independent sets $S$, and that correlations of real-valued multiplicative functions along weakly independent, jointly equidistributed shifts factorize as a product of single-function logarithmic means.","pith_inferences":["If the disjointness theorem behind Theorem 3.1 were available for Cesàro averages, the same transfer would upgrade Theorem 1.1 from logarithmic to unweighted averages, giving density-type vanishing along deterministic sequences.","The paper's Problem 1—that all joint correlations of $f_j\\circ a$ coincide with those of $f_j$—would, if true, imply that the partial sums of $f(a(n))$ diverge for every strongly aperiodic multiplicative $f$, and would remove the Sarnak-type obstruction described in Section 1.6.","A natural next test is to relax total ergodicity of $a$ to plain ergodicity; the paper's identity fails there, and the expected failure mode is a periodic obstruction, so one might look for counterexamples among deterministic sequences whose associated systems have irrational spectrum.","The factorization in Theorem 1.5 suggests that for real-valued multiplicative functions, logarithmic correlations ignore the fine structure of the shifts beyond congruence-equidistribution, a property that fails for complex-valued functions such as $n^{it}$ and $n^{-it}$, and this may be a useful probe for higher-order correlations beyond the currently accessible cases."],"forward_implications":["For $a(n)=[n\\alpha+\\beta]$ with irrational $\\alpha>1$ and distinct $n_1,n_2$, the logarithmically averaged two-point correlation of the Liouville function along $a$ is $0$: $E^{\\log}_{m\\in\\mathbb{N}}\\lambda(a(m+n_1))\\lambda(a(m+n_2))=0$.","For odd $\\ell$, the $\\ell$-point correlations of $\\lambda$ along every deterministic totally ergodic sequence vanish for distinct shifts, extending the odd-order vanishing results to composed sequences.","Sign patterns of a $\\{-1,1\\}$-valued multiplicative function that is not weakly pretentious to a Dirichlet character appear along deterministic totally ergodic sequences with the expected frequencies: triple patterns have logarithmic density $1/8$, and quadruple patterns have positive lower density.","For shift families with different growth rates or rationally independent Beatty frequencies, correlations with at least one strongly aperiodic factor vanish as $|n|\\to\\infty$, and all $2^{\\ell+1}$ sign patterns of the Liouville function occur along the shifts for all but finitely many $n$.","For real-valued multiplicative functions along weakly independent, jointly equidistributed shifts, the logarithmic correlation equals the product of the individual logarithmic means, so the functions behave asymptotically like independent random variables under those shifts."],"supporting_citations":[{"why":"Provides the disjointness statement (Theorem 3.3) that all F-systems of bounded multiplicative functions are disjoint from zero-entropy totally ergodic systems.","marker":"[7]"},{"why":"Supplies the structural results on F-systems of multiplicative functions and the correlation identities behind Theorem 4.3.","marker":"[8]"},{"why":"Supplies the two-point vanishing theorem used in the ℓ=2 case of Theorem 1.1 and the entropy decrement argument that underlies the structural results.","marker":"[23]"},{"why":"Supplies the general vanishing of logarithmically averaged correlations when the product of the functions is not weakly pretentious to a Dirichlet character, used in all other cases of Theorem 1.1.","marker":"[26]"},{"why":"Gives the averaged-form result and the input that strongly aperiodic functions are orthogonal to the Kronecker factor of their F-system.","marker":"[22]"},{"why":"Supplies the constancy of means of real-valued multiplicative functions on typical short intervals used in Theorem 1.5.","marker":"[21]"},{"why":"Provides the theory of characteristic factors and nilsystems used to reduce Propositions 4.4 and 4.5 to nilsystem statements.","marker":"[18]"},{"why":"Proves that factors of order k are inverse limits of k-step nilsystems, enabling the induction over nilsystems.","marker":"[17]"}],"fun_headline_variants":["Multiplicative correlations vanish on deterministic shifts","Deterministic shifts kill multiplicative correlations","Ergodic proof: multiplicative correlations vanish on deterministic sequences","Correlation identity for multiplicative functions along deterministic sequences","Vanishing correlations for multiplicative functions on deterministic orbits"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof depends on a previously proved theorem, quoted without reproof, saying that the measure-preserving systems associated with bounded multiplicative functions are disjoint from every zero-entropy totally ergodic system; this theorem is currently established only for logarithmic averages, and if it failed in the stated generality the main conclusion would collapse.","fun_headline_variants_meta":{"raw":{"variants":["Multiplicative correlations vanish on deterministic shifts","Deterministic shifts kill multiplicative correlations","Ergodic proof: multiplicative correlations vanish on deterministic sequences","Correlation identity for multiplicative functions along deterministic sequences","Vanishing correlations for multiplicative functions on deterministic orbits"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000781,"raw_usage":{"total_tokens":3418,"prompt_tokens":882,"completion_tokens":2536,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":498,"completion_tokens_details":{"reasoning_tokens":2466}},"tokens_in":498,"tokens_out":2536,"duration_ms":18567,"temperature":1.0,"reasoning_tokens":2466,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:35:19.738952+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the logarithmic average of $\\lambda([m\\alpha])\\lambda([(m+1)\\alpha])$ for an irrational $\\alpha>1$; the theorem predicts the limit is $0$, so a persistent non-zero limit would disprove Theorem 1.1. A more conceptual test: exhibit any bounded multiplicative function whose associated measure-preserving system has non-trivial irrational spectrum (the paper's own candidate is the system $(x,y)\\mapsto(x,y+x)$ on $\\mathbb{T}^2$), which would break the quoted disjointness theorem and, for $a(n)=[n\\alpha]$, put $e^{2\\pi i\\alpha}$ in the spectrum of the composed sequence.","supporting_citations":[{"cited_title":"Frantzikinakis, B","cited_arxiv_id":null,"evidence_quote":"Provides the disjointness statement (Theorem 3.3) that all F-systems of bounded multiplicative functions are disjoint from zero-entropy totally ergodic systems."},{"cited_title":"Furstenberg systems of bounded multiplicative functions and applications","cited_arxiv_id":"1804.08556","evidence_quote":"Supplies the structural results on F-systems of multiplicative functions and the correlation identities behind Theorem 4.3."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the two-point vanishing theorem used in the ℓ=2 case of Theorem 1.1 and the entropy decrement argument that underlies the structural results."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the general vanishing of logarithmically averaged correlations when the product of the functions is not weakly pretentious to a Dirichlet character, used in all other cases of Theorem 1.1."},{"cited_title":"Matomäki, M","cited_arxiv_id":null,"evidence_quote":"Gives the averaged-form result and the input that strongly aperiodic functions are orthogonal to the Kronecker factor of their F-system."},{"cited_title":"Matomäki, M","cited_arxiv_id":null,"evidence_quote":"Supplies the constancy of means of real-valued multiplicative functions on typical short intervals used in Theorem 1.5."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the theory of characteristic factors and nilsystems used to reduce Propositions 4.4 and 4.5 to nilsystem statements."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Proves that factors of order k are inverse limits of k-step nilsystems, enabling the induction over nilsystems."}],"review_version":1}