{"id":"55be72bb-cefd-48eb-9540-688bd7b9fc5c","arxiv_id":"2507.22831","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":8.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"For any homogeneous linear equation over F_p, every solution-free set whose Cayley graph has sublinear independence number has sublinear size exactly when some nonempty subset of the coefficients sums to zero.","lead":"This paper proves a complete classification of when a homogeneous linear equation over a prime field has all its 'structured' solution-free sets small. It is a Ramsey-Turán style variant of Roth's theorem, linking additive combinatorics with extremal graph theory.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Most load-bearing step is the black-box use of Roth's theorem with distinct elements for zero-sum subequations; it is standard via supersaturation, but should be verified explicitly.","rationale":"The reader identified the external distinct-elements form of Roth's theorem as the weakest assumption, and I agree: Theorem 3.1 depends on it in a way that cannot be relaxed without modifying the proof, and the paper does not explicitly prove or cite the supersaturation version. However, the standard Fourier-analytic proof of Roth's theorem for zero-sum equations immediately yields the stronger statement: the number of solutions in a set of density ε is about ε^{k'}p^{k'-1}, while degenerate solutions with repeated coordinates lie in lower-dimensional subspaces and number only O(p^{k'-2}), so distinct solutions exist for all sufficiently large p. The same supersaturation argument even supports the iterative deletion step, since the count remains positive after losing O(1) elements. I also checked the rest of the proof: the rainbow-path lemma and its application are internally consistent, the non-degenerate construction is correct, and the quantitative lower bounds, aside from a minor miscount in the bound on |XΣ| in Theorem 4.4 (r q^r should be about r q^{2r}), are valid and easily patched without changing the polynomial exponent. No other load-bearing gaps surfaced, so the reader's ACCEPT verdict stands.","tokens_in":14836,"tokens_out":52177,"duration_ms":594443,"concrete_test":"For the minimal zero-sum subset S used in Theorem 3.1, take the subequation L' and prove the supersaturation bound: for any A⊆F_p with |A|≥εp, the number of solutions to L' in A is ≥(ε^{k'}-o(1))p^{k'-1}, while the number with x_i=x_j for some i<j is ≤C p^{k'-2}; combining these gives a solution with all k' coordinates distinct for all sufficiently large p. Run this check for the worst case k'=3 with coefficients (1,1,-2) and for an arbitrary zero-sum equation.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 3.1 derives the entire degenerate direction and the quantitative bound d(L,ε) ≤ 100^{k+1}k^3ε from repeated applications of Theorem 1.1 to a zero-sum subequation L' with k'≥3 variables. Each application must return a solution whose coordinates are distinct and disjoint from previously deleted elements; if Theorem 1.1 only guaranteed nontrivial (not necessarily distinct) solutions, then the auxiliary solution family S in the proof of Theorem 3.1 would not satisfy the disjointness condition C2, and the subsequent rainbow-path construction could produce repeated z_i. This is a genuine load-bearing external assumption. In the standard proof, however, the number of all solutions in a set of size εp is (ε^{k'}+o(1))p^{k'-1}, while solutions with at least one pair of equal coordinates are only O(p^{k'-2}) (or O(ε^{k'-2}p^{k'-2})), so for sufficiently large p a distinct solution always exists. Thus the assumption is true but is not stated or proved in the manuscript; the paper would be more robust if it cited or derived the supersaturation form explicitly.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves a Ramsey-Turán analogue of Roth's theorem for homogeneous linear equations over F_p. The main result (Theorem 1.2) classifies equations for which every solution-free set A with independence number α(Cay_{F_p}(A))=o(p) must have size o(p): this holds exactly when some nonempty subset of the coefficients sums to zero. The degenerate direction is proved with the quantitative bound d(L,ε) ≤ 100^{k+1} k^3 ε, using a new rainbow directed path lemma combined with Roth's theorem; the non-degenerate direction is witnessed by an explicit construction with |A| ≥ βp and α(Cay(A)) = O_L(p/log p). The paper also establishes d(L,ε)=Θ(ε) for Schur's equation and d(L,ε)=ε^{Θ(1)} for every non-degenerate equation possessing a proper zero-sum subset of coefficients.","tokens_in":80,"tokens_out":54429,"duration_ms":1287694,"significance":"If the results hold, the paper gives a clean classification of density-regular homogeneous equations under the Ramsey-Turán independence condition, a natural structural restriction introduced by Erdős and Sárközy. The proof strategy is original: the degenerate case is reduced to a graph-theoretic rainbow path lemma, while the non-degenerate case is handled by explicit constructions that transfer Ramsey graph lower bounds into the additive setting. The paper is careful about the distinct-elements convention in the definition of solution-free sets, and the main classification proof is detailed and self-contained apart from standard tools (Roth's theorem, Caro-Wei, Ramsey graph lower bounds). The quantitative bounds, though probably not optimal except for Schur's equation, give a clear picture of the ε-dependence and raise interesting open problems.","major_comments":[{"comment":"The proof claims that a graph G with q=t^k vertices and α(G)≤t^{k−1} is guaranteed by Theorem 4.3. With n=t^k, Theorem 4.3 only gives α(G)≤C t^{k−1} log t; the extra log factor invalidates the pigeonhole step α(Cay_Fp(X))≤p/t, because q/t is no longer larger than α(G). The theorem is likely repairable by choosing n≈t^k polylog(t) and absorbing the resulting polylog factors into the exponent C, but as written the proof of d(L,ε)≥ε^C is incomplete.","section":"Section 4, Theorem 4.4"}],"minor_comments":[{"comment":"In the induction step, the second and fourth bullets refer to 'the color of →v_i v_{i+1} in D_{i+1}' but the edge from v_i to v_{i+1} in the proper rainbow path lies in D_i; the index should be D_i.","section":"Section 3.1, Lemma 3.2"},{"comment":"In the proof of the solution-free claim, the sentence 'assume y_1 is the largest among the y_i's' should read 'assume z_1 is the largest among the z_i's'.","section":"Section 3.2, Theorem 3.3"},{"comment":"The displayed inclusion 'X_i±X_i ⊆ (8p/9,p) ∪ [0,p/9)' is not literally correct: X_i+X_i is contained in [0,2p/9), while X_i−X_i is contained in (8p/9,p) ∪ [0,p/9). The argument only needs both sets to be disjoint from Y=[p/3,4p/9], so the proof is unaffected, but the inclusion should be stated accurately.","section":"Section 4, Theorem 4.2"},{"comment":"The base-r digit argument is written as if the pairs {a_j,b_j} are distinct, but coefficients c_j can repeat or cancel; the footnote about partial cancellation should be expanded so that the 'each exponent appears in at least two distinct pairs' conclusion is fully justified.","section":"Section 4, Theorem 4.4"},{"comment":"The bound 'at most r q^r' for the size of X_Σ appears too small; a safe bound such as (2|X|)^r would still lead to a polynomial bound for r' after absorbing constants, so the numerical estimate should be corrected or loosened.","section":"Section 4, Theorem 4.4"},{"comment":"Several lemmas are referred to as theorems (e.g., 'Theorem 2.1', 'Theorem 2.2', 'Theorem 3.2' for Lemma 2.1, Lemma 2.2, and Lemma 3.2); consistent numbering would avoid confusion.","section":"Throughout"},{"comment":"The repeated application of Roth's theorem to obtain the family S of disjoint solutions is correct because Theorem 1.1 is stated for the distinct-elements notion of solution-free, but a sentence making this contrapositive explicit would improve readability.","section":"Section 3.1, Theorem 3.1"}],"recommendation":"major_revision","confidential_remarks":"The main classification theorem (Theorem 1.2) appears correct and well-proved; the only substantive issue I found is the gap in Theorem 4.4's use of Theorem 4.3, which affects the lower bound in Theorem 1.4. I believe this is repairable by enlarging the auxiliary graph by polylog factors and absorbing them into the exponent of ε. I therefore recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper does something genuinely new: it takes the Ramsey–Turán idea of forbidding large independent sets in the Cayley graph and applies it to Roth's density-regularity theorem for homogeneous linear equations over F_p. The main result, an iff classification in terms of a zero-sum subset of coefficients, is absent from the cited literature and is proved in both directions. The quantitative definitions d(L,ε) and the bounds (Theorem 3.1 and the Θ(ε) result for Schur's equation) are useful and likely to get cited.\n\nWhat the paper does well: the graph-theoretic core, especially the proper rainbow path lemma (Lemma 3.2), is clean and the greedy argument works. The non-degenerate direction gives an explicit construction with |A| ≥ βp and independence number O(p/log p), which is exactly the kind of structured example that makes the classification interesting. The Schur lower bound uses a Ramsey graph in a standard but careful way, and the general polynomial lower bound via Osthus–Taraz is fine. The proof of the degenerate direction applies Roth's theorem repeatedly to extract many disjoint solutions of a zero-sum subequation, then builds a rainbow path on the remaining coordinates. I checked the counting: after deleting the coordinates of ℓ solutions, there is still enough left to keep applying Roth, so the iteration is legitimate.\n\nThe stress-test note worried that the black-box use of Roth's theorem might only give nontrivial solutions with equal coordinates, not distinct ones. That concern does not land. The paper's Theorem 1.1 is stated with the same distinct-elements convention used throughout: a solution-free set is defined by forbidding k-tuples of distinct elements. So the contrapositive gives distinct solutions directly, and the repeated deletion preserves disjointness. The note's suggested supersaturation remark would be a nice addition for expository completeness, but it is not needed for correctness.\n\nSoft spots are minor: a few internal cross-references are off (e.g., references to Theorem 2.2 where Lemma 2.2 is meant), and there is a typo 'y_1' for 'z_1' in the proof of Theorem 3.1. None of these affect the arguments. The paper does not oversell itself; the concluding remarks honestly discuss limitations for other groups and the open quantitative questions.\n\nWho should read this: anyone working in additive combinatorics or extremal graph theory, especially on Ramsey–Turán problems or density regularity of equations. It deserves a serious referee. I would be happy to cite it and would bring it to the reading group.","headline":"A clean, genuinely new Ramsey–Turán variant of Roth's theorem over F_p with a complete classification; the proof is solid and the small blemishes are typo-level.","tokens_in":15577,"tokens_out":3356,"would_cite":true,"duration_ms":38244,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05D10","11B30","05C25","05C35"],"pacs":[],"model":"deepseek-v4-flash","headline":"For a homogeneous linear equation over F_p, solution-free sets with sublinear Cayley independence are small exactly when a nonempty subset of the coefficients sums to zero, completing the Ramsey–Turán analogue of Roth's theorem.","keywords":["Ramsey–Turán theory","Roth's theorem","density regularity","homogeneous linear equations","Cayley graphs","independence number","finite fields","additive combinatorics"],"falsifier":"Take the Schur equation $x+y-z=0$ and fix a small $\\varepsilon>0$; Theorem 3.1 says $\\limsup_{p\\to\\infty}D(L,\\varepsilon,p)/p\\le 100^{4}\\cdot27\\,\\varepsilon$, while Theorem 1.3 says the true order is $\\Theta(\\varepsilon)$. Any infinite family of primes whose normalized maximum size exceeds that upper bound, or any degenerate equation whose normalized maximum size decays more slowly than linearly in $\\varepsilon$, would contradict the quantitative claims; for the non-degenerate direction, checking the paper's explicit construction for $x+y+z=0$ on moderate primes is a finite verification that its independence is $O(p/\\log p)$.","tokens_in":14672,"feed_emoji":"🔢","tokens_out":13548,"duration_ms":149024,"temperature":0.7,"pith_summary":"This paper asks when a subset $A\\subset\\mathbb{F}_p$ that contains no solution to a fixed homogeneous linear equation $L$ must be small, after one rules out the usual structured extremal examples by demanding that the Cayley graph generated by $A$ have independence number $o(p)$. The answer is a complete classification: every such solution-free set has size $o(p)$ if and only if some nonempty set of $L$'s coefficients sums to zero. This is a Ramsey–Turán variant of the classical density-regularity theorem for homogeneous linear equations, adding the sublinear-independence condition that the classical extremal examples fail. The degenerate case comes with the quantitative bound $d(L,\\varepsilon)\\le 100^{k+1}k^3\\varepsilon$, and the paper shows this linear dependency is tight for the Schur equation and polynomial in general.","feed_headline":"Zero-sum coefficient subsets decide solution density in F_p","feed_subtitle":"A Ramsey–Turán variant of Roth's theorem pins the threshold on whether coefficients add to zero.","key_machinery":"The engine is a lemma about directed graph systems. A restricted digraph system consists of properly edge-colored digraphs on a common vertex set together with a function assigning each vertex a small set of forbidden colors; a proper rainbow directed path is a path whose $i$-th edge lies in the $i$-th digraph, whose edge colors are all distinct, and whose colors avoid the forbidden sets of the two endpoints. The lemma shows that when each digraph has independence number at most $|V|/(100^{k'}\\ell^2)$, such a path of length $k'$ must exist. In the degenerate case the Cayley digraphs generated by coefficient dilations $c_iA$ are the digraphs, colors are the elements of $A$, and the zero-sum subset of coefficients provides a disjoint family of solutions of a smaller subequation; the rainbow path then threads these solutions together into a full solution of $L$.","core_discovery":"Let $L:c_1x_1+\\cdots+c_kx_k=0$ be a homogeneous linear equation with $k\\ge 3$ nonzero integer coefficients, and call $A\\subseteq\\mathbb{F}_p$ solution-free when no $k$-tuple of distinct elements of $A$ solves $L$. The theorem states that every solution-free $A$ with $\\alpha(\\mathrm{Cay}_{\\mathbb{F}_p}(A))=o(p)$ has $|A|=o(p)$ exactly when $L$ is degenerate, meaning some nonempty subset of its coefficients sums to zero. For non-degenerate equations the paper constructs solution-free sets of linear size whose Cayley graphs still have independence $O(p/\\log p)$, so sublinear independence does not force smallness. For degenerate equations the proof yields $d(L,\\varepsilon)\\le 100^{k+1}k^3\\varepsilon$, where $d(L,\\varepsilon)$ is the asymptotic maximum density of solution-free sets with independence at most $\\varepsilon p$.","pith_inferences":["The same classification is plausibly portable to $\\mathbb{Z}$ with a difference-intersector condition; the paper notes the translation is routine, so an exact integer analogue would be a direct check rather than a new theorem.","The quantitative gap between $O(\\varepsilon)$ and $\\varepsilon^{\\Theta(1)}$ suggests testing intermediate degenerate equations: equations with an isolated zero-sum pair may already force the linear rate, while equations needing larger zero-sum subsets may not.","If the independence assumption is strengthened to $o(p/\\log p)$ or smaller, the paper's non-degenerate constructions fail, so the classification boundary could shift; whether it does is an open question the paper explicitly raises.","The use of sparse high-girth graphs in the lower bound suggests that progress in Ramsey–Turán graph constructions would translate directly into sharper polynomial exponents for non-degenerate equations."],"forward_implications":["For any degenerate equation, the maximum density of a solution-free set with $\\alpha(\\mathrm{Cay}_{\\mathbb{F}_p}(A))\\le\\varepsilon p$ is at most $100^{k+1}k^3\\varepsilon$, so the density tends to zero with $\\varepsilon$.","For any non-degenerate equation, there is a fixed $\\beta>0$ and a solution-free set of size at least $\\beta p$ with $\\alpha=O(p/\\log p)$, which is $o(p)$; hence sublinear independence does not suffice to force smallness unless a zero-sum coefficient subset exists.","For the Schur equation $x+y-z=0$, the quantitative rate is $d(L,\\varepsilon)=\\Theta(\\varepsilon)$, so the general linear upper bound cannot be improved in the exponent.","For every equation whose coefficients have nonzero total sum but some zero-sum subset, the lower bound is $\\varepsilon^{\\Theta(1)}$, matching the qualitative classification.","A set with $\\alpha(\\mathrm{Cay}_{\\mathbb{F}_p}(A))=o(p)$ is a difference intersector, meeting $B-B$ for every $B$ of positive linear density; the theorem therefore classifies when every difference-intersecting solution-free set is small."],"supporting_citations":[{"why":"Supplies the density theorem for zero-sum homogeneous equations that the degenerate-case proof invokes repeatedly.","marker":"[17]"},{"why":"Gives the independence-number lower bound in terms of average degree used in the preliminary graph lemmas.","marker":"[5]"},{"why":"Cited jointly with [5] for the same independence-number bound.","marker":"[24]"},{"why":"Introduces the Ramsey–Turán type condition of imposing sublinear independence that motivates the whole problem.","marker":"[8]"},{"why":"Introduces the difference-intersector notion of structured sets underlying the Cayley-graph condition.","marker":"[7]"},{"why":"Gives the triangle-free graph lower bound used to construct Schur-solution-free sets.","marker":"[13]"},{"why":"Supplies the high-girth graph with small independence number used in the general polynomial lower bound.","marker":"[15]"}],"fun_headline_variants":["Zero-sum coefficient subsets decide solution-free density","Degenerate equations force small sets in Roth-type problem","Ramsey–Turán Roth tied to zero-sum coefficient subsets","Sublinear independence only helps for zero-sum equations","A zero-sum subset of coefficients makes solution-free sets tiny"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The degenerate direction leans on the classical density theorem that every constant-positive-density subset of $\\mathbb{F}_p$ contains a solution to any zero-sum equation in at least three variables with distinct entries; if that theorem or its distinct-entry convention failed, the classification would not follow.","fun_headline_variants_meta":{"raw":{"variants":["Zero-sum coefficient subsets decide solution-free density","Degenerate equations force small sets in Roth-type problem","Ramsey–Turán Roth tied to zero-sum coefficient subsets","Sublinear independence only helps for zero-sum equations","A zero-sum subset of coefficients makes solution-free sets tiny"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000236,"raw_usage":{"total_tokens":1486,"prompt_tokens":907,"completion_tokens":579,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":523,"completion_tokens_details":{"reasoning_tokens":503}},"tokens_in":523,"tokens_out":579,"duration_ms":6548,"temperature":1.0,"reasoning_tokens":503,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T11:19:18.255517+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the Schur equation $x+y-z=0$ and fix a small $\\varepsilon>0$; Theorem 3.1 says $\\limsup_{p\\to\\infty}D(L,\\varepsilon,p)/p\\le 100^{4}\\cdot27\\,\\varepsilon$, while Theorem 1.3 says the true order is $\\Theta(\\varepsilon)$. Any infinite family of primes whose normalized maximum size exceeds that upper bound, or any degenerate equation whose normalized maximum size decays more slowly than linearly in $\\varepsilon$, would contradict the quantitative claims; for the non-degenerate direction, checking the paper's explicit construction for $x+y+z=0$ on moderate primes is a finite verification that its independence is $O(p/\\log p)$.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the density theorem for zero-sum homogeneous equations that the degenerate-case proof invokes repeatedly."},{"cited_title":"Caro,New results on the independence number, Tech","cited_arxiv_id":null,"evidence_quote":"Gives the independence-number lower bound in terms of average degree used in the preliminary graph lemmas."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Cited jointly with [5] for the same independence-number bound."},{"cited_title":"Erd˝ os and V","cited_arxiv_id":null,"evidence_quote":"Introduces the Ramsey–Turán type condition of imposing sublinear independence that motivates the whole problem."},{"cited_title":"Erd˝ os and A","cited_arxiv_id":null,"evidence_quote":"Introduces the difference-intersector notion of structured sets underlying the Cayley-graph condition."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the triangle-free graph lower bound used to construct Schur-solution-free sets."},{"cited_title":"Osthus and A","cited_arxiv_id":null,"evidence_quote":"Supplies the high-girth graph with small independence number used in the general polynomial lower bound."}],"review_version":1}