Optimal Lifting for the Projective Action of $SL_3(Z)$

Amitay Kamber; Hagai Lavner

REVIEW 1 major objections 5 minor 19 references

Optimal Lifting for the Projective Action of $SL_3(Z)$

T0 review · 1 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict New higher-rank lifting theorem with a real but likely fixable trace identity error; worth a careful referee. read the letter →

arxiv 1908.06682 v3 pith:ZQIYJ4HF submitted 2019-08-19 math.NT

Amitay Kamber , Hagai Lavner This is my paper · ORCID

classification math.NT MSC 11F0611F7222E4011P21

keywords optimalstrongapproximationprojectiveplaneoverfinitefieldSL(3Z)spectralgaplatticepointcountingcongruencesubgroupsflagvarietiesHarish-Chandrafunction

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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).

What would settle it

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.

Watch

Extended reading notes

Core claim

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 $\Gamma'_0(q)$.

Load-bearing premise

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 $q$.

Editorial extensions

If this is right

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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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 $1/2$.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Dave,

This 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}.

What'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.

The 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).

The 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.

The 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.

My 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.

Referee Report

1 major / 5 minor

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.

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 (1)

[§3, Eq. (3.2)] 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.

minor comments (5)

[§3, Eq. (3.1)] 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.
[§5, Lemma 5.5] 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.
[§4, displayed equation after 'So it suffices to show that'] 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.
[§4, Lemma 4.7 proof] 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.
[§4, Theorem 4.3] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the lifting theorem is derived from an elementary counting estimate plus an external uniform spectral-gap input, and the optimal exponent is a volume lower bound.

full rationale

The derivation chain is self-contained rather than circular. Theorem 1.3 is obtained from Theorem 1.4 (the counting input) by the analytic argument in Section 4; Theorem 1.4 is proved in Section 3 by direct counting over SL_3(Z) with divisor bounds, with no use of the theorem it feeds. The only deep spectral input, Theorem 4.3, is cited to [6, Section 4] (Ghosh-Gorodnik-Nevo), an external, parameter-free statement whose assumptions do not include the target lifting result, so the citation is genuine evidence rather than a self-referential load. The paper's own self-citation [8] is invoked only as a conceptual framework and for discussion; the actual bound used, including the uniform spectral gap and Harish-Chandra bounds, comes from external sources. The optimality of the exponent 1/3 is a volume comparison (|P_q| ~ q^2 versus ~T^6 matrices of norm at most T), not an a posteriori fit, and it is independent of the proof of the upper bound. I find no equation in the paper where an output quantity is defined in terms of an input quantity or where a fitted parameter is renamed as a prediction. The apparent algebraic inconsistency in Eq. (3.2) noted in the surrounding discussion is a correctness risk, not a circularity: even if that identity were erroneous, the counting argument would be invalid rather than circular, and the failure mode would be an internal error, not a reduction of the claim to its own assumptions.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No fitted parameters; the exponent is forced by the volume comparison. The proof relies on standard external theorems (uniform spectral gap, property (T), Harish-Chandra bounds) which are not derived in the paper.

assumptions (5)

domain assumption Uniform spectral gap for Γ_0(q)\G/K (Theorem 4.3): exists τ>0 such that ||f*χ_T||_2 ≪ T^{-τ}||f||_2 for all f∈L^2_0(X_q), uniformly in q.
Cited to [6, Section 4]; this is the bridge from the elementary counting theorem to the lifting theorem and is not proved in the paper.
domain assumption Explicit property (T) for SL3(R) with uniform Kazhdan constants for all lattices.
Underlies Theorem 4.3; cited to Oh [15].
standard math Haar measure asymptotics for balls in SL3(R): μ({||g||_K ≤ T}) ≍ T^6 and μ({||g||_δ ≤ T}) ≍ log(T) T^2.
Used to normalize convolution kernels and to justify the optimal exponent; cited to [13, 14].
standard math Harish-Chandra Ξ function bounds (4.2): ||g||_δ^{-1} ≤ Ξ(g) ≪ (log||g||)^{C0} ||g||_δ^{-1}.
Key to the convolution lemma; cited to [19, 2.1].
standard math Counting Theorem 2.1 for SL2 congruence subgroups (from [5, Lemma 5.3]).
Used in the SL2 model proof; not needed for the SL3 theorem but part of the exposition.

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Cite this review

Pith. "Pith review of Optimal Lifting for the Projective Action of $SL_3(Z)$." pith.science (2026). https://pith.science/paper/ZQIYJ4HF

@misc{pith2026190806682,
  author       = {Pith},
  title        = {Pith review of: Optimal Lifting for the Projective Action of $SL_3(Z)$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZQIYJ4HF}},
  note         = {Machine review of arXiv:1908.06682}
}

abstract

Let $\epsilon>0$ and let $q$ be a prime going to infinity. We prove that with high probability given $x,y$ in the projective plane over the finite field $F_q$ there exists $\gamma$ in $SL_3(Z)$, with coordinates bounded by $q^{1/3+\epsilon}$, whose projection to $SL_{3}(F_q)$ sends $x$ to $y$. The exponent $1/3$ is optimal and the result is a high rank generalization of Sarnak's optimal strong approximation theorem for $SL_2(Z)$.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

19 extracted references · 17 canonical work pages

[8]

On Sarnak’s density conjecture and its applications

Golubev, K., and Kamber, A. On Sarnak’s density conjecture and its applications. arXiv preprint arXiv:2004.00373 (2020). 1, 1

arXiv 2020
[6]

Diophantine approximation and automorphic spectrum

Ghosh, A., Gorodnik, A., and Nevo, A. Diophantine approximation and automorphic spectrum. Interna- tional Mathematics Research Notices 2013 , 21 (2013), 5002–5058. 2, 4, 4

work page 2013
[1]

Density theorems for GL(n)

Blomer, V. Density theorems for GL(n). arXiv preprint arXiv:1906.07459 (2019). 1

arXiv 2019
[2]

Applications of the Kuznetsov formula on GL(3) II: the level aspect

Blomer, V., Buttcane, J., and Maga, P. Applications of the Kuznetsov formula on GL(3) II: the level aspect. Mathematische Annalen 369 , 1-2 (2017), 723–759. 1

work page 2017
[3]

Ramanujan duals and automorphic spectrum

Burger, M., Li, J.-S., and Sarnak, P. Ramanujan duals and automorphic spectrum. Bulletin of the American Mathematical Society 26 , 2 (1992), 253–257. 1

work page 1992
[4]

Density of integer points on aﬃne homogeneous varieties

Duke, W., Rudnick, Z., and Sarnak, P. Density of integer points on aﬃne homogeneous varieties. Duke mathematical journal 71 , 1 (1993), 143–179. 1

work page 1993
[5]

congruence

Gamburd, A. On the spectral gap for inﬁnite index "congruence" subgroup s of SL2(Z). Israel Journal of Mathematics 127 , 1 (2002), 157–200. 1, 2, 2

work page 2002
[7]

Best possible rates of distribution of dense lattice orbits in homo- geneous spaces

Ghosh, A., Gorodnik, A., and Nevo, A. Best possible rates of distribution of dense lattice orbits in homo- geneous spaces. Journal für die reine und angewandte Mathematik (Crelles Jo urnal) (2014). 1

work page 2014

Show all 19 references

[9]

H., and Wright, E

Hardy, G. H., and Wright, E. M. An introduction to the theory of numbers . Oxford university press, 1979. 2

1979
[10]

Hea th, T. C. On a Spectral Bound for Congruence Subgroup Families in SL(3, Z). PhD thesis, Columbia University, 2015. 5

2015
[11]

Density theorems for congruence groups in real rank 1

Huntley, J., and Ka tznelson, Y. Density theorems for congruence groups in real rank 1. Duke Math. J 71 (1993), 463–473. 1

1993
[12]

Huxley, M. N. Exceptional eigenvalues and congruence subgroups. The Selberg Trace Formula and Related Topics, Contemp. Math 53 (1986), 341–349. 1, 2

1986
[13]

Knapp, A. W. Representation Theory of Semisimple Groups: An Overview Ba sed on Examples (PMS-36) . Princeton university press, 2016. 4

2016
[14]

Homogeneous asymptotic limits of haar measures of semisimp le linear groups and their lattices

Maucourant, F. Homogeneous asymptotic limits of haar measures of semisimp le linear groups and their lattices. Duke Mathematical Journal 136 , 2 (2007), 357–399. 1, 1, 4

2007
[15]

Uniform pointwise bounds for matrix coeﬃcients of unitary r epresentations and applications to Kazhdan constants

Oh, H. Uniform pointwise bounds for matrix coeﬃcients of unitary r epresentations and applications to Kazhdan constants. Duke mathematical journal 113 , 1 (2002), 133–192. 1

2002
[16]

Notes on the generalized Ramanujan conjectures

Sarnak, P. Notes on the generalized Ramanujan conjectures. Harmonic analysis, the trace formula, and Shimura varieties 4 (2005), 659–685. 2

2005
[17]

Letter to Stephen D

Sarnak, P. Letter to Stephen D. Miller and Naser Talebizadeh Sardari on optimal strong approximation by integral points on quadrics. 2015. 1, 1

2015
[18]

Sarnak, P., and Xue, X. X. Bounds for multiplicities of automorphic representations . Duke Math. J 64 , 1 (1991), 207–227. 1, 1, 2

1991
[19]

C., and V aradarajan, V

Trombi, P. C., and V aradarajan, V. S. Asymptotic behaviour of eigen functions on a semisimple Lie group: the discrete spectrum. Acta Mathematica 129 , 1 (1972), 237–280. 4

1972

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [8]

On Sarnak’s density conjecture and its applications

Golubev, K., and Kamber, A. On Sarnak’s density conjecture and its applications. arXiv preprint arXiv:2004.00373 (2020). 1, 1

arXiv 2020

[2] [6]

Diophantine approximation and automorphic spectrum

Ghosh, A., Gorodnik, A., and Nevo, A. Diophantine approximation and automorphic spectrum. Interna- tional Mathematics Research Notices 2013 , 21 (2013), 5002–5058. 2, 4, 4

work page 2013

[3] [1]

Density theorems for GL(n)

Blomer, V. Density theorems for GL(n). arXiv preprint arXiv:1906.07459 (2019). 1

arXiv 2019

[4] [2]

Applications of the Kuznetsov formula on GL(3) II: the level aspect

Blomer, V., Buttcane, J., and Maga, P. Applications of the Kuznetsov formula on GL(3) II: the level aspect. Mathematische Annalen 369 , 1-2 (2017), 723–759. 1

work page 2017

[5] [3]

Ramanujan duals and automorphic spectrum

Burger, M., Li, J.-S., and Sarnak, P. Ramanujan duals and automorphic spectrum. Bulletin of the American Mathematical Society 26 , 2 (1992), 253–257. 1

work page 1992

[6] [4]

Density of integer points on aﬃne homogeneous varieties

Duke, W., Rudnick, Z., and Sarnak, P. Density of integer points on aﬃne homogeneous varieties. Duke mathematical journal 71 , 1 (1993), 143–179. 1

work page 1993

[7] [5]

congruence

Gamburd, A. On the spectral gap for inﬁnite index "congruence" subgroup s of SL2(Z). Israel Journal of Mathematics 127 , 1 (2002), 157–200. 1, 2, 2

work page 2002

[8] [7]

Best possible rates of distribution of dense lattice orbits in homo- geneous spaces

Ghosh, A., Gorodnik, A., and Nevo, A. Best possible rates of distribution of dense lattice orbits in homo- geneous spaces. Journal für die reine und angewandte Mathematik (Crelles Jo urnal) (2014). 1

work page 2014