{"id":"89077495-ea07-4f64-acbd-fbefc7d23eb9","arxiv_id":"1908.06682","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For almost all pairs of points in P^2(F_q), an element of SL3(Z) with entries bounded by q^{1/3+ε} realizes the projective action.","lead":"The authors prove that for almost every pair of points in the projective plane over a large prime field, there is an integral 3 by 3 matrix with entries bounded by q^{1/3+ε} that maps the first point to the second. This attains the optimal exponent 1/3 and generalizes Sarnak's optimal strong approximation theorem from SL2 to SL3.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof of Theorem 1.4 relies on an incorrect trace identity, Eq. (3.2), for bad matrices; the central counting step is formally invalid until corrected.","rationale":"The reader's weakest_assumption was the uniform spectral gap, Theorem 4.3, but the reader's rationale also flagged the trace identity error in Eq. (3.2). I agree with the reading that the paper is likely correct after minor repairs, and I share the CONDITIONAL verdict. The spectral gap is cited to a standard source and follows from explicit property (T) for SL3(R), so it is not the most concrete soft spot in the text. The trace identity error is an actual internal inconsistency in the proof of Theorem 1.4, the counting theorem on which the main theorem rests. The argument would be valid if Eq. (3.2) is corrected to trγ = 2α + α^{-2} and trγ^{-1} = 2α^{-1} + α^2, but as printed the proof of Theorem 1.4 is incomplete. The attack is not an attack on the authors or on the plausibility of the result; it is a precise, checkable failure in the written argument.","tokens_in":21498,"tokens_out":25536,"duration_ms":267220,"concrete_test":"Re-derive Section 3's trace relations from the characteristic polynomial. For a matrix with eigenvalues α, α, α^{-2} and determinant 1, verify that the characteristic polynomial is x^3 - (2α+α^{-2})x^2 + (2α^{-1}+α^2)x - 1. Replace the printed Eq. (3.2) with the correct identities and re-run the S,R dyadic count for bad γ, including the exceptional cases and Lemma 5.5. If the corrected identities still give the bounds S^3(R/q+1)^2 ≪ RSq and the Lemma 5.5 count ≪ (S/q+1)(R/q+1)+q, the central claim is repairable; if any step changes materially, Theorem 1.4 is unsupported.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"In Section 3, a matrix γ that is bad mod q is defined to have an eigenvalue α with a two-dimensional eigenspace and third eigenvalue α^{-2}. Its eigenvalues are therefore α, α, α^{-2}, and the characteristic polynomial forces trγ = 2α + α^{-2} and trγ^{-1} = 2α^{-1} + α^2. The text instead prints trγ = α + 2α^{-2} and trγ^{-1} = α^{-1} + 2α^2. This is not a harmless relabelling: the printed identities are inconsistent with the preceding sentence and with Eq. (3.3). Substituting the printed traces into (3.3) gives α^2(α+2α^{-2}) - α(α^{-1}+2α^2) = 1 - α^3, whereas the required identity is α^3 - 1. The incorrect (3.2) is then used in the proof of Theorem 1.4 to conclude that α is one of at most three roots of a known cubic and to determine trγ^{-1}; the same equations underlie Lemma 5.5. As printed, the derivation of the crucial counting estimate for bad γ does not go through. This is an internal inconsistency in the elementary counting argument, not merely a missing citation. It is very likely a typo in the multiplicity convention, but it is load-bearing because Theorem 1.4 feeds directly into the analytic proof of Theorem 1.3.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves an optimal-lifting theorem for the projective action of SL3(Z) on P^2(F_q): for every ε>0 and for almost all pairs (x,y) in P^2(F_q), there exists γ ∈ SL3(Z) with ||γ||_∞ ≤ q^{1/3+ε} whose reduction mod q sends x to y. The exponent 1/3 is shown to be optimal. The proof combines an elementary counting theorem (Theorem 1.4) bounding the number of pairs (γ,x) with γ fixing x in P^2(F_q), with a spectral argument based on property (T) and a uniform spectral gap for the quotients Γ_0(q)\\SL3(R)/SO(3). A flag-variety analogue (Theorem 5.1) is also proved with the optimal exponent 1/2. The paper's main new ingredient is the self-contained counting argument in Section 3, while Section 4 follows the Sarnak–Xue strategy as adapted in prior work of one of the authors.","tokens_in":21804,"tokens_out":14932,"duration_ms":124244,"significance":"If the proof is corrected, this is a substantial higher-rank analogue of Sarnak's optimal strong approximation theorem, and it appears to be the first optimal result of this type for a projective action of SL3. The elementary counting theorem is self-contained and gives explicit bounds of the form q^{2+ε}T; the optimality of the exponent 1/3 follows correctly from comparing the ball count (≈T^6) with the size of P^2(F_q) (≈q^2). The flag-variety theorem is a nontrivial extension. The analytic part is standard but relies on an external uniform spectral gap, which the paper cites from [6]. The main reservation is the incorrect trace identity in Section 3, which is load-bearing for the proof of Theorem 1.4 but appears to be a typo with a straightforward correction.","major_comments":[{"comment":"The trace formula in Eq. (3.2) is false for the eigenvalue data stated. A matrix over F_q with a two-dimensional α-eigenspace and third eigenvalue α^{-2} has eigenvalues (α, α, α^{-2}), hence trγ = 2α + α^{-2} and trγ^{-1} = 2α^{-1} + α^2 mod q. The displayed identities trγ = α + 2α^{-2} and trγ^{-1} = α^{-1} + 2α^2 correspond to the different type (α, α^{-2}, α^{-2}). Substituting the printed traces into Eq. (3.3) gives α^2(α+2α^{-2}) - α(α^{-1}+2α^2) = 1-α^3, rather than α^3-1, so the printed system is internally inconsistent. Since Eqs. (3.2) and (3.3) are used in the proof of Theorem 1.4 to determine α and trγ^{-1}, the printed derivation of the bad-matrix count does not go through. The inconsistency also surfaces in the exceptional-case analysis, where the statement that the trace of the lower 2×2 block is 'either 0 or 2' matches the correct trace formula but not the printed one. This appears to be a typo: replacing (3.2) by the correct traces makes (3.3) an identity and preserves the subsequent 'at most three options for α' and the recovery of trγ^{-1}. The authors should correct Eq. (3.2) and re-verify all steps in Section 3 and in Lemma 5.5 that depend on it.","section":"§3, Eq. (3.2)"}],"minor_comments":[{"comment":"The right-hand side of Eq. (3.1) should be written as (α + α^{-2})I, since it is a scalar matrix identity; the current text '= α + α^{-2} mod q' is slightly ambiguous.","section":"§3, Eq. (3.1)"},{"comment":"In the proof of Lemma 5.5, the definitions of z and w appear to have mismatched subscripts: for two solutions (x1,y1,α) and (x2,y2,α), one needs z=(x1-x2)/q and w=(y1-y2)/q for the reduction to Eq. (5.1) to follow from Eq. (3.3). The printed definitions z=(x1-y1)/q and w=(x2-y2)/q make the displayed implication invalid as written.","section":"§5, Lemma 5.5"},{"comment":"The displayed summation in the proof of Lemma 4.5 contains a typographical error: the condition should be ||x_0^{-1}γx_0||_δ ≤ C_2 q^2, not ||x_0^{-1}γx_0||_δ^{-1} ≤ C_2 q^2. As printed, the summation condition is inconsistent with the preceding and following lines.","section":"§4, displayed equation after 'So it suffices to show that'"},{"comment":"In the last sentence of the proof of Lemma 4.7, 'which says that x0γ = y' should read 'x0g = y', since the element constructed in the argument is denoted g.","section":"§4, Lemma 4.7 proof"},{"comment":"Theorem 4.3 is a crucial spectral input for Theorem 1.3, and the manuscript cites [6, Section 4] with the comment that the result holds uniformly for all lattices in SL3(R). Since the uniformity over the particular non-compact family Γ_0(q) is essential, the authors should quote the precise theorem from [6] (or the relevant statement from [8]) so the reader can verify that the assumption is exactly what is needed.","section":"§4, Theorem 4.3"}],"recommendation":"major_revision","confidential_remarks":"The error in Eq. (3.2) is almost certainly typographical, but it sits at the base of the main counting theorem, so the revision should include a careful re-verification of all steps that depend on the trace identities. The paper also relies on a uniform spectral gap for non-compact quotients cited to [6]; the editor may want the authors to spell out the quoted statement. The self-citation [8] for the analytic framework should be made available or its relevant parts summarized. The paper otherwise fits the journal's scope and, if corrected, makes a solid contribution."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dave,\n\nThis one is worth reading, but don't take Section 3 as printed. The new theorem is real: first optimal strong approximation for a higher-rank lattice action on a non-compact homogeneous space, with exponent 1/3 for SL3(Z) mod p acting on P^2. The strategy is coherent—elementary counting of fixed points plus a uniform spectral gap—and the volume comparison makes the exponent sharp: ~q^2 points versus ~T^6 matrices, so T~q^{1/3}.\n\nWhat's good: the paper proves Theorem 1.4 with a self-contained elementary argument (except the spectral gap cited to Ghosh–Gorodnik–Nevo), and the flag generalization in Theorem 5.1 is a natural next step. The introduction is honest about not proving Conjecture 1.2 for the principal subgroup.\n\nThe soft spot is load-bearing. Eq (3.2) prints trγ = α+2α^{-2}, trγ^{-1} = α^{-1}+2α^2. For a bad γ with eigenvalues α, α, α^{-2}, the correct traces are 2α+α^{-2} and 2α^{-1}+α^2. Plugging the printed values into (3.3) gives the negative of the required identity; the derivation of \"at most three options for α\" and the bound on bad γ does not go through as written. This is almost certainly a typo—the preceding sentence and the required (3.3) both point to the corrected identity—but it is not cosmetic: it blocks the proof of Theorem 1.4 until fixed. The same typo infects Lemma 5.5, which cites (3.2) and (3.3).\n\nThe other soft spot is Theorem 5.1: the analytic part is \"left to the reader\" and the Jordan-form classification in Section 5 is also delegated. For a paper whose main theorem is new, that is acceptable only as a sketch, not as a full proof.\n\nThe uniform spectral gap, Theorem 4.3, is cited to Ghosh–Gorodnik–Nevo rather than proved; if that citation holds, the deduction works. The self-citation [8] is not a problem here—it supplies the framework, not the result.\n\nMy recommendation: engage. Send it to a serious referee. The result is likely true, the flaw is fixable, and the payoff is worth it. I would want the revision to correct (3.2), re-verify Lemma 5.5, and either prove or explicitly delegate the flag case. It deserves referee time.","headline":"New higher-rank lifting theorem with a real but likely fixable trace identity error; worth a careful referee.","tokens_in":22348,"tokens_out":3712,"would_cite":true,"duration_ms":34762,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11F06","11F72","22E40","11P21"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that for almost every pair of points in the projective plane over a finite field, a matrix in $SL_3(\\mathbb{Z})$ with entries bounded by $q^{1/3+\\epsilon}$ sends one to the other, and the exponent $1/3$ is optimal.","keywords":["optimal strong approximation","projective plane over finite field","SL(3,Z)","spectral gap","lattice point counting","congruence subgroups","flag varieties","Harish-Chandra function"],"falsifier":"Compute, for a fixed prime $q$ (say $q=101$), the set of pairs $(x,y)\\in P^2(\\mathbb{F}_q)^2$ that are connected by some $\\gamma\\in SL_3(\\mathbb{Z})$ with $\\|\\gamma\\|_\\infty \\le q^{1/3}$. If this set does not have density $1-o(1)$ in $P_q^2$, the central theorem fails at that scale; if the density is very close to 1 only for $q\\ge q_0$, the uniformity in $q$ can be checked. A separate check targets Theorem 1.4 directly: enumerate all pairs $(\\gamma,x)$ with $\\|\\gamma\\|_\\infty\\|\\gamma^{-1}\\|_\\infty\\le C q^2$ and $\\gamma x=x$; if the count exceeds a constant times $q^{2+\\epsilon}$ for any fixed epsilon, the counting lemma is wrong.","tokens_in":21286,"feed_emoji":"🔢","tokens_out":10029,"duration_ms":88645,"temperature":0.7,"pith_summary":"The paper proves an optimal strong approximation theorem for the action of the integer matrix group $SL_3(\\mathbb{Z})$ on the projective plane over a finite field. For every $\\epsilon>0$ and all large primes $q$, almost every point $x$ in the projective plane over $\\mathbb{F}_q$ has the property that almost every other point $y$ can be reached from $x$ by a matrix with entries bounded by $q^{1/3+\\epsilon}$. The exponent $1/3$ cannot be improved, because there are about $q^2$ projective points while only about $T^6$ integer matrices have entries of size at most $T$. This is the first high-rank analogue of a known optimal lifting result for $SL_2(\\mathbb{Z})$, and it is obtained by combining an elementary counting bound with a spectral gap that holds uniformly as $q$ varies.","feed_headline":"Cube-root integer matrices reach almost every projective pair","feed_subtitle":"Optimal lifting for SL(3,Z): the exponent 1/3 cannot be improved.","key_machinery":"The proof has two load-bearing parts. The first is an elementary counting theorem (Theorem 1.4): the number of pairs $(\\gamma,x)$ with $\\gamma\\in SL_3(\\mathbb{Z})$, $x\\in P^2(\\mathbb{F}_q)$, $\\gamma x=x$, and $\\|\\gamma\\|_\\infty \\|\\gamma^{-1}\\|_\\infty \\le T$ is $O_\\epsilon(q^{2+\\epsilon}T)$ for $T\\le C q^2$. This is proved by classifying matrices that are 'bad' modulo $q$ (those with a repeated eigenvalue) and counting their entries using determinant-trace relations and divisor bounds. The second is a uniform spectral gap (Theorem 4.3): for the quotients $X_q=\\Gamma_0(q)\\backslash SL_3(\\mathbb{R})/SO(3)$ and the normalized characteristic function $\\chi_T$ of a $K$-ball of radius $T$, the convolution operator $f\\mapsto f*\\chi_T$ on the zero-integral subspace satisfies $\\|f*\\chi_T\\|_2 \\ll T^{-\\tau}\\|f\\|_2$ with a fixed $\\tau>0$ independent of $q$. The analytic section converts the counting bound into an almost-everywhere statement using normalized point-pair functions $b_{T,x}$, a convolution lemma based on Harish-Chandra's $\\Xi$ function, and property (T).","core_discovery":"The central result, Theorem 1.3, states that for every $\\epsilon>0$, as $q\\to\\infty$ among primes, there is a set $Y\\subset P_q = P^2(\\mathbb{F}_q)$ of size at least $(1-o_\\epsilon(1))|P_q|$ such that for each $x\\in Y$ there is a set $Z_x\\subset P_q$ of size at least $(1-o_\\epsilon(1))|P_q|$ such that for every $y\\in Z_x$ there is some $\\gamma\\in SL_3(\\mathbb{Z})$ with $\\|\\gamma\\|_\\infty \\le q^{1/3+\\epsilon}$ and $\\Phi_q(\\gamma)x=y$. In other words, for all but $o_\\epsilon(|P_q|^2)$ pairs $(x,y)$, a matrix of bounded size $q^{1/3+\\epsilon}$ realizes the projective transformation. The exponent $1/3$ is optimal: the number of matrices with $\\|\\gamma\\|_\\infty\\le T$ grows like $T^6$, matching the $q^2$ points of the projective plane only when $T=q^{1/3}$, and the special point $x=(0,0,1)$ in fact requires matrices of size at least $q^{2/3}$ to reach most targets. The theorem is an 'on average' version of optimal strong approximation for the non-principal congruence subgroup $\\Gamma'_0(q)$.","pith_inferences":["The elementary counting mechanism is likely to extend to $SL_N(\\mathbb{Z})$ acting on partial flag varieties; the predicted optimal exponent for a variety of dimension $d$ would be $d/(N^2-N)$, with the same 'bad matrix' classification becoming a classification of semisimple versus unipotent elements modulo $q$.","Because the uniform spectral gap appears as an external input, the paper provides a template: any family of congruence quotients with a uniform mean ergodic theorem and a matching count of fixed points will yield an optimal lifting statement, even when the full Ramanujan conjectures are false.","A natural testable extension is the action of $SL_3(\\mathbb{Z})$ on the space of $k$-dimensional subspaces of $\\mathbb{F}_q^3$ for $k=1$ and $k=2$; the authors' result for flags suggests the exponent should interpolate between $1/3$ and $1/2$."],"forward_implications":["For every $\\epsilon>0$, as $q\\to\\infty$, the exceptional set of pairs $(x,y)\\in P_q^2$ that cannot be joined by a matrix of norm $\\le q^{1/3+\\epsilon}$ has size $o_\\epsilon(q^4)$.","No exponent smaller than $1/3$ can work: for any $\\delta>0$, the number of matrices with norm $\\le q^{1/3-\\delta}$ is too small relative to the $q^2$ points of the projective plane, and the point $(0,0,1)$ requires norm at least $q^{2/3}$ to reach most targets.","The same strategy proves an analogous optimal result for complete flags in $\\mathbb{F}_q^3$, with the optimal exponent $1/2$ (Theorem 5.1).","The special point $(0,0,1)$ shows the average statement is genuinely about generic points: matrices of size $q^{2/3+\\epsilon}$ already suffice to reach the entire projective plane from that point."],"supporting_citations":[{"why":"Provides the SL_2 optimal strong approximation theorem that this paper generalizes to the projective action.","marker":"[17]"},{"why":"Supplies the uniform spectral gap (Theorem 4.3) that converts the counting bound into an almost-everywhere lifting statement.","marker":"[6]"},{"why":"Gives the Haar measure asymptotics of balls (size ~T^6 and T^2 log T) that determine the optimal exponents.","marker":"[14]"},{"why":"Introduces the lattice-point-counting approach to bounding exceptional eigenvalues that this paper adapts.","marker":"[18]"},{"why":"Provides the Cartan decomposition integration formula used to normalize the convolution kernels.","marker":"[13]"},{"why":"Supplies the bounds for Harish-Chandra's Xi function used in the convolution lemma.","marker":"[19]"},{"why":"Gives asymptotics for the number of integer matrices in balls, used to set the optimal exponent.","marker":"[4]"},{"why":"Contains a counting lemma for SL_2(Z) used in the warm-up proof.","marker":"[5]"}],"fun_headline_variants":["Optimal cube-root lifting for SL(3,Z) actions","Almost all projective pairs lift with cube-root matrices","SL(3,Z) reaches most pairs with cube-root matrices","Exponent 1/3 is optimal for SL(3,Z) projective lifting","Optimal on-average strong approximation for SL(3,Z)"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The entire argument rests on a uniform spectral gap: the rate at which convolutions with large balls damp out the zero-mean part of a function on the quotient $\\Gamma_0(q)\\backslash SL_3(\\mathbb{R})/SO(3)$ must stay positive and independent of the prime $q$.","fun_headline_variants_meta":{"raw":{"variants":["Optimal cube-root lifting for SL(3,Z) actions","Almost all projective pairs lift with cube-root matrices","SL(3,Z) reaches most pairs with cube-root matrices","Exponent 1/3 is optimal for SL(3,Z) projective lifting","Optimal on-average strong approximation for SL(3,Z)"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000604,"raw_usage":{"total_tokens":2812,"prompt_tokens":934,"completion_tokens":1878,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":550,"completion_tokens_details":{"reasoning_tokens":1792}},"tokens_in":550,"tokens_out":1878,"duration_ms":14343,"temperature":1.0,"reasoning_tokens":1792,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:42:18.594872+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute, for a fixed prime $q$ (say $q=101$), the set of pairs $(x,y)\\in P^2(\\mathbb{F}_q)^2$ that are connected by some $\\gamma\\in SL_3(\\mathbb{Z})$ with $\\|\\gamma\\|_\\infty \\le q^{1/3}$. If this set does not have density $1-o(1)$ in $P_q^2$, the central theorem fails at that scale; if the density is very close to 1 only for $q\\ge q_0$, the uniformity in $q$ can be checked. A separate check targets Theorem 1.4 directly: enumerate all pairs $(\\gamma,x)$ with $\\|\\gamma\\|_\\infty\\|\\gamma^{-1}\\|_\\infty\\le C q^2$ and $\\gamma x=x$; if the count exceeds a constant times $q^{2+\\epsilon}$ for any fixed epsilon, the counting lemma is wrong.","supporting_citations":[{"cited_title":"Letter to Stephen D","cited_arxiv_id":null,"evidence_quote":"Provides the SL_2 optimal strong approximation theorem that this paper generalizes to the projective action."},{"cited_title":"Diophantine approximation and automorphic spectrum","cited_arxiv_id":null,"evidence_quote":"Supplies the uniform spectral gap (Theorem 4.3) that converts the counting bound into an almost-everywhere lifting statement."},{"cited_title":"Homogeneous asymptotic limits of haar measures of semisimp le linear groups and their lattices","cited_arxiv_id":null,"evidence_quote":"Gives the Haar measure asymptotics of balls (size ~T^6 and T^2 log T) that determine the optimal exponents."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the lattice-point-counting approach to bounding exceptional eigenvalues that this paper adapts."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the Cartan decomposition integration formula used to normalize the convolution kernels."},{"cited_title":"C., and V aradarajan, V","cited_arxiv_id":null,"evidence_quote":"Supplies the bounds for Harish-Chandra's Xi function used in the convolution lemma."},{"cited_title":"Density of integer points on affine homogeneous varieties","cited_arxiv_id":null,"evidence_quote":"Gives asymptotics for the number of integer matrices in balls, used to set the optimal exponent."},{"cited_title":"congruence","cited_arxiv_id":null,"evidence_quote":"Contains a counting lemma for SL_2(Z) used in the warm-up proof."}],"review_version":1}