{"id":"a5373d3e-941b-452b-8acc-5a718b4f64b2","arxiv_id":"1908.02611","paper_version":2,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Invariant measures on the n-torus that survive a sparse family of shifted power maps must be finitely supported or Lebesgue, provided the shifts form a strongly independent tuple.","lead":"This mathematics paper introduces strongly independent matrices and proves a rigidity theorem for measures on the n-dimensional torus: any invariant measure that also stays invariant under a sparse family of related matrix maps must be either finite or the uniform Lebesgue measure. The result extends known one-dimensional rigidity to higher dimensions using a clean linear algebra condition.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified; the load-bearing Fourier characterization (Theorem 4.1) and the rigidity proof in §5 are sound under scrutiny.","rationale":"The reader correctly identified Theorem 4.1 and its reliance on Lemma 4.5 as the load-bearing premise. My stress-test confirms that this is the critical step but finds it correct. The proof of Theorem 4.1 is internally consistent, the extension from trigonometric polynomials to L^2 is justified by uniform boundedness after invariance is established, and the use of strong independence in Lemma 4.4 is sound. The rigidity argument in Section 5 then follows exactly as stated, with no hidden assumptions about invertibility of A or tempering of Følner sequences. Therefore no verdict adjustment is needed.","tokens_in":11061,"tokens_out":44139,"duration_ms":417181,"concrete_test":"Independently verify Lemma 4.5 for a non-invertible endomorphism by computing the L^2-norm of Følner averages on the unilateral shift part and the exponential-sum bound on the unitary part; if either estimate fails for some Følner sequence, Theorem 4.1 and hence Theorem 1.7 would require qualification.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim rests on Theorem 4.1 and its dependence on the mean ergodic theorem (Lemma 4.5) for arbitrary Følner sequences and possibly non-invertible A. I examined this premise. For a measure-preserving, possibly non-invertible map, the Koopman operator is an isometry on L^2; its Wold decomposition splits into a unitary part and a unilateral shift part. On the unitary part, any Følner set has o(|F|) maximal intervals, so the exponential sums vanish at every nontrivial character. On the shift part, the average of shifted finite-support vectors has L^2-norm O(1/|F|), and a density argument gives strong convergence. Thus Lemma 4.5 is valid, and the extension from characters to all of L^2 in Theorem 4.1 uses only the uniform bound ||(1/|F|)∑ f∘T^j||_2 ≤ ||f||_2. I found no internal inconsistency or missing case in the proof of Theorem 1.7: the density arguments in cases (1)–(3), Lemma 4.4, and the finite-support/Dirac conclusion all follow as written.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces the notion of strongly independent n-tuples of matrices over a field F: for every nonzero column vector v, the vectors B1v,...,Bnv are linearly independent. It proves Theorem 1.5, that a nonzero matrix is strongly independent exactly when its characteristic polynomial is irreducible, and Theorem 1.6, that no strongly independent n-tuple exists over an algebraically closed field for n≥2. The main dynamical result is Theorem 1.7: if an ergodic, weakly mixing, or strongly mixing ×A-invariant Borel probability measure on T^n is additionally invariant under ×(A^j+B_i) for all i and for a density-one, positive-density, or infinite set of exponents j, with (B1,...,Bn) strongly independent over Q, then the measure is either finitely supported or Lebesgue; in the weakly and strongly mixing cases it is either Dirac or Lebesgue. The proof proceeds through a Fourier-coefficient characterization of ergodicity, weak mixing, and strong mixing for ×A-invariant measures (Theorem 4.1) and then uses Lemma 4.4 to force finite support from the extra invariances. A corollary produces small abelian semigroups of integer matrices for which Lebesgue measure is the unique non-atomic invariant measure.","tokens_in":11268,"tokens_out":16004,"duration_ms":165043,"significance":"If the results hold, Theorem 1.7 is a substantial higher-dimensional analogue of the first author's one-dimensional rigidity theorem, and the algebraic part is elegant: the equivalence with irreducibility gives an abundant supply of admissible tuples over Q, while Theorem 1.6 shows the obstruction over algebraically closed fields. The Fourier characterization is stated in useful generality, including possibly non-invertible integer matrices and arbitrary Følner sequences, and the rigidity argument is concise and correct. The paper is carefully organized and the main hypotheses are sharp in their density/strength distinctions. The proof is essentially self-contained apart from two quoted lemmas from [H], namely Lemma 4.5 (mean ergodic theorem for amenable semigroups) and Lemma 5.1 (weakly mixing measures with an atom are Dirac); these are used as black boxes rather than as assumptions of the rigidity conclusion, so I do not see circularity. Overall this is a solid contribution with a clean new algebraic condition replacing the scalar non-lacunarity condition.","major_comments":[],"minor_comments":[{"comment":"The passage from the identity (4.5) on trigonometric polynomials to all f,g in L2(T^n,μ) is asserted in one sentence. Please add the standard approximation argument: after proving μ is T_A-invariant, one has ||f∘T_A^j||_2 = ||f||_2 for every j, and this uniform bound allows a density argument. As written the step is correct but too compressed.","section":"§4, proof of Theorem 4.1(1)"},{"comment":"The same compressed density extension appears in the sufficiency direction: from (4.3) for characters to f,g in L2. Please spell out the approximation with the uniform L2 bounds, since this is what turns the Fourier identities into the mixing condition for Borel sets.","section":"§4, proof of Theorem 4.1(3)"},{"comment":"When applying part (1) to μ×μ to prove ergodicity of the product, the text does not explicitly state that μ×μ is invariant under T_{diag(A,A)}; this follows from the T_A-invariance of μ and should be said before invoking part (1).","section":"§4, proof of Theorem 4.1(2)"},{"comment":"The assertion that the homogeneous polynomial f(z)=det[B1z ... Bnz] has a nonzero zero because F is algebraically closed and n≥2 deserves a brief justification: a nonzero homogeneous polynomial of positive degree in at least two variables has a nontrivial projective zero over an algebraically closed field, and if f is identically zero the conclusion is immediate.","section":"§3, proof of Theorem 1.6"},{"comment":"The notation B_i in the definition of the semigroup S is ambiguous. If B_i is meant to be B^i with B0=In, this should be stated explicitly, so that the strongly independent tuple is (B0,B1,...,B^{n-1}).","section":"§5, proof of Corollary 1.8"},{"comment":"After passing to a subsequence of the Følner sequence to make |Fm∩E|/|Fm| tend to 1, the proof should note explicitly that the subsequence is still a Følner sequence, so that Theorem 4.1 can be applied to it.","section":"§5, proof of Theorem 1.7(1)"},{"comment":"There are several typographical slips, including 'm ultiplicative' in §1 and 'Adv anced' in reference [Ro]; these should be corrected in a final pass.","section":"Throughout"}],"recommendation":"minor_revision","confidential_remarks":"I found no circularity and no sign of parameter fitting; the dependence on [H] is limited to two cited lemmas that function as standard black boxes. The paper is a solid extension of the first author's earlier one-dimensional rigidity result and the algebraic criterion is clean and well matched to the dynamical application."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper does what it says: it proves measure rigidity on T^n for endomorphisms with sparse semigroups of shifts, extending Huang's one-dimensional result and giving a new route into Furstenberg-type questions. The new notion of strongly independent matrices is the right abstraction, and the main theorem is a real step beyond the circle case.\n\nWhat is actually new: (1) The characterization that a matrix is strongly independent exactly when its characteristic polynomial is irreducible (Theorem 1.5). This is useful and clean. (2) The non-existence of strongly independent tuples over algebraically closed fields (Theorem 1.6). (3) The rigidity theorem itself, Theorem 1.7, which handles ergodic, weakly mixing, and strongly mixing ×A-invariant measures on T^n and concludes either finite support (or a Dirac, in mixing cases) or Lebesgue. This is a substantive result, not a repackaging of known work.\n\nI checked the main line of proof. The Fourier characterization in Section 4 is standard and proved carefully; the L2 approximation steps are compressed but routine. The rigidity argument in Section 5 uses the Fourier identities correctly: the extra invariances force \\hat{\\mu}(B_i k)=1 for some nonzero k, and the strong independence of the B_i's then forces finite support via Lemma 4.4. That works.\n\nSoft spots: the paper leans on two quoted results, Lemma 4.5 (mean ergodic theorem for amenable semigroups, credited to Bewley) and Lemma 5.1 (atomic weakly mixing implies Dirac, from Huang). Both are standard, and the first is proved in the cited literature. A referee might want the arguments to be more self-contained, but this is not a substantive gap. The density arguments in cases (1)--(3) of Theorem 1.7 are terse; a slightly fuller write-up would help readers less familiar with Følner density arguments. Nothing here looks wrong.\n\nWho this is for: ergodic theorists working on Furstenberg-type measure rigidity, especially people interested in higher-dimensional generalizations. The linear algebra part (Section 3) is also of independent interest. The paper deserves a serious referee; I would accept it for a strong dynamics journal, with minor suggestions for expanding the compressed steps. It does not resolve the full Furstenberg conjecture, but that is not a flaw.","headline":"A genuine n-dimensional measure rigidity result with clean proofs; the main theorem appears correct and the paper deserves serious ref.","tokens_in":11785,"tokens_out":1208,"would_cite":true,"duration_ms":13140,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37A05","37A25","37A46","43A05","28C10","12E05"],"pacs":[],"model":"deepseek-v4-flash","headline":"Strongly independent perturbations force torus-invariant measures to be atomic or Lebesgue.","keywords":["strongly independent matrices","measure rigidity","n-torus","Fourier coefficients","ergodic theory","weak mixing","strong mixing","Furstenberg conjecture"],"falsifier":"Exhibit an ergodic $\\times A$-invariant probability measure on $\\mathbb{T}^n$ that is non-atomic, not Lebesgue, and satisfies the extra $\\times(A^j+B_i)$-invariance for a strongly independent over $\\mathbb{Q}$ tuple for $j$ in a set of Følner upper density one; the theorem says no such measure exists, and the proof predicts one would find $\\hat{\\mu}(B_i k)\\neq 1$ for some nonzero $k$ despite the hypotheses.","tokens_in":10858,"feed_emoji":"🔁","tokens_out":10558,"duration_ms":94577,"temperature":0.7,"pith_summary":"The paper establishes a measure rigidity theorem for endomorphisms of the $n$-torus: an ergodic, weakly mixing, or strongly mixing probability measure that is invariant under the map $x \\mapsto Ax$ and also under the maps $x \\mapsto (A^j+B_i)x$ for a family of integer matrices $B_1,\\dots,B_n$ that are strongly independent over $\\mathbb{Q}$ must be either finitely supported (a Dirac measure in the mixing cases) or the Lebesgue measure. To make this work, the authors introduce a purely algebraic notion: a tuple of matrices is strongly independent if no nontrivial linear combination maps any nonzero vector to zero, and they prove a single matrix is strongly independent exactly when its characteristic polynomial is irreducible, yielding abundant examples over $\\mathbb{Q}$. A complementary result shows no such tuples exist over algebraically closed fields for $n\\ge 2$. The proof runs through a Fourier-coefficient characterization of ergodicity and mixing for $\\times A$-invariant measures, which turns the extra invariances into the equations $\\hat{\\mu}(B_i k)=1$ and then forces the support to be finite.","feed_headline":"Strongly independent perturbations trivialize torus measures","feed_subtitle":"New rigidity theorem: only atomic or Lebesgue measures survive on the n-torus under these conditions.","key_machinery":"The central object is the notion of a strongly independent tuple of matrices over a field $F$: an $n$-tuple $(B_1,\\dots,B_n)$ such that for every nonzero column vector $v$, the vectors $B_1v,\\dots,B_nv$ are linearly independent, or equivalently every nontrivial linear combination $\\sum u_jB_j$ is invertible. For a single matrix $B$, this property holds exactly when the characteristic polynomial of $B$ is irreducible in $F[t]$ (Theorem 1.5), so companion matrices of irreducible integer polynomials provide strongly independent tuples over $\\mathbb{Q}$. The other load-bearing mechanism is Theorem 4.1, which characterizes ergodicity, weak mixing, and strong mixing of a $\\times A$-invariant measure purely by the limiting behaviour of the Fourier coefficients $\\hat{\\mu}(A^j k+l)$ along a Følner sequence; this lets the extra invariances be read directly as coefficient identities and, combined with strong independence, forces the finite-support conclusion via Lemma 4.4.","core_discovery":"The central claim is that the one-dimensional rigidity of $\\times p$-invariant measures under $\\times(p^j+l)$-invariance, previously established in [H], extends to the full $n$-torus once the perturbation $l$ is replaced by a matrix $B_i$ and the tuple $(B_1,\\dots,B_n)$ is strongly independent over $\\mathbb{Q}$. Stated on its own terms: for a fixed $A\\in M_n(\\mathbb{Z})$, the family of maps $\\times(A^j+B_i)$ for $j$ in a large set (Følner upper density one, positive density, or infinite, depending on the mixing assumption) is enough to rigidify the $\\times A$-action. If the invariant measure is not Lebesgue, there is a nonzero frequency $k$ with $\\hat{\\mu}(k)\\neq 0$, and the Fourier characterization of Theorem 4.1 forces $\\hat{\\mu}(B_i k)=1$ for every $i$. Strong independence then implies the vectors $B_1 k,\\dots,B_n k$ are linearly independent, so the support of $\\mu$ is contained in a finite set of torus points, giving the dichotomy.","pith_inferences":["The proof of Theorem 1.7 only uses strong independence of the tuple at the single nonzero frequency $k$; one could weaken the hypothesis to 'for every nonzero $k$ with $\\hat{\\mu}(k)\\neq 0$, the vectors $B_1 k,\\dots,B_n k$ are linearly independent over $\\mathbb{Q}$' and the same finite-support conclusion would hold, a frequency-wise version of the condition.","The non-existence result over algebraically closed fields (Theorem 1.6) suggests the rigidity is inherently arithmetic: over $\\mathbb{C}$ there is no algebraic way to force such a rich perturbation family, so the phenomenon depends on the ground field.","A similar strategy might prove rigidity for actions on other compact abelian groups (e.g., solenoids) where Fourier coefficients and exact endomorphisms make sense, using the same strong-independence hypothesis on the Pontryagin dual."],"forward_implications":["Corollary 1.8 follows: there exists an abelian semigroup $S\\subseteq M_n(\\mathbb{Z})$ for which the Lebesgue measure is the unique non-atomic measure on $\\mathbb{T}^n$ invariant under every $\\times A$ with $A\\in S$ and ergodic under some individual $\\times B$.","The Fourier-coefficient characterization of Theorem 4.1 supplies an iff test for mixing of $\\times A$-invariant measures that works for any integer matrix $A$ and any Følner sequence, not just for automorphisms.","In the weakly and strongly mixing cases, the finite-support alternative strengthens to a Dirac measure (Lemma 5.1), so the dichotomy in those cases is atomic versus Lebesgue.","Because any matrix with irreducible characteristic polynomial is strongly independent (Theorem 1.5), the theorem provides many explicit families of perturbation matrices for which rigidity holds, requiring no Diophantine or lacunarity conditions."],"supporting_citations":[{"why":"Supplies the mean ergodic theorem for amenable semigroups (Lemma 4.5) used to derive the Fourier-coefficient characterization.","marker":"[B]"},{"why":"Provides the one-dimensional rigidity theorem being generalized, the Fourier-coefficient method behind Theorem 4.1, and Lemma 5.1 for reducing finite support to Dirac in mixing cases.","marker":"[H]"},{"why":"Provides the standard fact (Lemma 4.2) that measure invariance under a continuous map is equivalent to invariance of all Fourier coefficients.","marker":"[W]"},{"why":"Supplies the Hamilton–Cayley theorem and Eisenstein's criterion used to prove the existence of strongly independent matrices (Theorem 1.5 and Remark 3.3).","marker":"[La]"}],"fun_headline_variants":["Strong independence collapses torus measures to trivial","Rigidity on n-torus: only atomic or Lebesgue survive","Independent matrix perturbations trivialize torus measures","Strongly independent matrices rigidify n-torus actions","Non-atomic torus measures vanish under strong independence"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole argument rests on the Fourier-coefficient characterization of Theorem 4.1—that Cesàro averages of $\\hat{\\mu}(A^j k+l)$ over a Følner sequence correctly detect ergodicity, weak mixing, and strong mixing for $\\times A$-invariant measures; if that test failed, the proof's step forcing $\\hat{\\mu}(B_i k)=1$ would collapse.","fun_headline_variants_meta":{"raw":{"variants":["Strong independence collapses torus measures to trivial","Rigidity on n-torus: only atomic or Lebesgue survive","Independent matrix perturbations trivialize torus measures","Strongly independent matrices rigidify n-torus actions","Non-atomic torus measures vanish under strong independence"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000406,"raw_usage":{"total_tokens":2053,"prompt_tokens":827,"completion_tokens":1226,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":443,"completion_tokens_details":{"reasoning_tokens":1149}},"tokens_in":443,"tokens_out":1226,"duration_ms":9734,"temperature":1.0,"reasoning_tokens":1149,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:59:43.039246+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Exhibit an ergodic $\\times A$-invariant probability measure on $\\mathbb{T}^n$ that is non-atomic, not Lebesgue, and satisfies the extra $\\times(A^j+B_i)$-invariance for a strongly independent over $\\mathbb{Q}$ tuple for $j$ in a set of Følner upper density one; the theorem says no such measure exists, and the proof predicts one would find $\\hat{\\mu}(B_i k)\\neq 1$ for some nonzero $k$ despite the hypotheses.","supporting_citations":[],"review_version":1}