Anisotropic quadratic equations in three variables

Jiamin Li; Jianya Liu

REVIEW 2 major objections 4 minor 23 references

Anisotropic quadratic equations in three variables

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Pith's one-line read An indefinite anisotropic ternary quadratic form with one integer solution has infinitely many almost-prime solutions, with six prime factors conditionally improving to five.

desk verdict Solid improvement on almost-prime bounds for anisotropic ternary quadratics, but Theorem 1.2 ships with an unproved two-variable equidistribution input. read the letter →

arxiv 2501.15033 v2 pith:U6D7N7JC submitted 2025-01-25 math.NT

Jiamin Li , Jianya Liu This is my paper · ORCID

classification math.NT MSC 11E2011N3611F72

keywords quadraticformalmost-primeweightedsieveSelbergeigenvalueconjectureanisotropicquadricspingroupZariskidensity

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

This paper proves that an indefinite anisotropic integral quadratic form in three variables, once it has one integer solution, has infinitely many integer solutions whose coordinates are almost-primes. There are infinitely many solutions with the first coordinate having at most 6 prime factors outside the fixed set {2,3,5,7}, and infinitely many with the product of the first two coordinates having at most 16 such prime factors. If Selberg's eigenvalue conjecture holds, those counts drop to 5 and 14. The point is an almost-prime substitute for the prime-variable problem on ternary quadrics, which is currently out of reach.

What carries the argument

The engine is an equidistribution estimate (Lemma 2.3) for the weighted number $a_n(T)$ of orbit points with first coordinate $\pm n$, summed over $n\equiv 0 \pmod d$: the sum equals $\omega(d)X/d + O_{\varepsilon}(d^{1+\varepsilon}T^{1/2+\theta+\varepsilon})$, with local density $\omega(p)/p=1/p+O(1/p^2)$. This gives the level of distribution $\tau = 1/4-\theta/2 = 25/128$ using the best unconditional bound $\theta=7/64$ toward Selberg's eigenvalue conjecture, and $\tau=1/4$ if the conjecture holds. That level feeds a one-dimensional linear sieve for Theorem 1.1 and a two-dimensional nonlinear sieve for Theorem 1.2, whose numerical optimization produces the constants 6 and 16.

What would settle it

Compute, for a concrete anisotropic form such as the norm form of a quaternion division algebra over $\mathbb{Q}$, the weighted counts $a_n(T)$ on a fixed orbit, and compare the left side of (2.8) with $\omega(d)X/d$ for many $d$ near $T^{25/128}$; a discrepancy exceeding the claimed $O_{\varepsilon}(d^{1+\varepsilon}T^{1/2+\theta+\varepsilon})$ would disprove the level of distribution on which the theorems rest.

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Extended reading notes

Core claim

Let $f(x_1,x_2,x_3)$ be an integral indefinite anisotropic quadratic form whose determinant $d(f)$ satisfies $td(f)$ square-free, and assume the congruence equation $f(x)\equiv t \pmod d$ is solvable for every $d$. The paper shows that the set of integer solutions with $x_1\in P_6(B)$ is Zariski dense in the affine quadric $V$, and likewise that $\{x: x_1x_2\in P_{16}(B)\}$ is Zariski dense; under Selberg's eigenvalue conjecture the constants improve to 5 and 14. Here $P_r(B)$ denotes integers with at most $r$ prime factors outside $B=\{2,3,5,7\}$. The proof combines the orbit structure under the spin double cover of the special orthogonal group preserving $f$, an equidistribution estimate for weighted counts of orbit points, and one- and two-dimensional weighted sieves, with spectral information coming from the quaternion algebra behind the form.

Load-bearing premise

The argument stands on the equidistribution estimate of Lemma 2.3: for all d up to T, the weighted solutions split across residue classes 0 modulo d with an error term of size $d^{1+\varepsilon} T^{1/2+\theta+\varepsilon}$; if that error were worse, the sieve would not deliver six or sixteen prime factors.

Editorial extensions

If this is right

For every indefinite anisotropic ternary form satisfying the hypotheses, one integral solution implies infinitely many with $x_1\in P_6(B)$, and the set of such solutions is Zariski dense rather than confined to a subvariety.
Infinitely many solutions have $x_1x_2\in P_{16}(B)$, so the almost-prime condition can be moved from all three coordinates to just one or two.
A proof of Selberg's eigenvalue conjecture would immediately lower the constants to 5 and 14.
Improving the unconditional spectral bound $\theta=7/64$ would raise the level $\tau$ and, through the same sieve inequalities, lower the constants before a full Selberg proof.
These results strengthen the earlier three-coordinate theorem $x_1x_2x_3\in P_{26}(B)$ (22 conditional) by showing that two or even one coordinate can be left unrestricted.

Reading between the lines

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

The paper does not compute the exact threshold in $\theta$ at which $r=5$ becomes unconditional; plugging improved spectral bounds into the sieve inequalities would yield such a threshold, showing how far current technology is from the unconditional five-prime-factor result.
Because the equidistribution input is stated for an arbitrary orbit of the spin group, the same sieve framework should apply to any ternary quadric whose integral points form finitely many orbits and whose attendant quaternion algebra supplies the spectral gap; other anisotropic forms are natural test cases.
The square-free condition on $td(f)$ keeps local densities close to $1/p$; locating where the proof uses this condition would indicate whether relaxing it merely worsens the constants or changes the qualitative conclusion.
A three- or higher-dimensional analogue for products of more than two coordinates would need a higher-dimensional weighted sieve, and the nonlinear sieve machinery in the paper is already set up to handle such extensions.

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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

Colleague,

Here's my read of Li–Liu, arXiv:2501.15033. The real new results are Theorem 1.1, getting P6 for a single coordinate on anisotropic ternary quadratics (P5 under Selberg), and Theorem 1.2 claiming P16 for a pair product (P14 under Selberg). Next to the prior P26 for x1x2x3 in Liu–Sarnak and P521 per coordinate in Blomer–Brüdtern, that is a genuine step. The 1-dimensional sieve for one coordinate is a genuinely new configuration, and the explicit sieve computations in §4 are careful and check out. This is a continuation of [18], but the new constants are outputs of the calculations, not inputs; no circularity.

The soft spot is Theorem 1.2. The paper never defines the sequence being sieved (the product x1x2), and never proves the required equidistribution modulo d for that product. Lemma 2.3 covers only the one-coordinate sequence with x1=±n. The condition d | x1x2 is a union of local conditions p|x1 or p|x2, and controlling the divisor sum needs a joint equidistribution statement for the pair (x1,x2). The sentence “the situation for two variables is similar” is not a proof. If the joint error term is worse than d^{1+ε}T^{1/2+θ+ε}, the level τ=25/128 used in §4.2 is unjustified, and the m(0.19214...)≈15.63 computation would not yield r=16. This is a missing proof obligation, not a demonstrated contradiction, and it is probably repairable with the same spectral machinery. But as written, Theorem 1.2 does not have a complete proof.

Smaller point: Lemma 2.3 is itself stated as “the proof is exactly the same as [18, Theorem 2.1]” after a short local-density argument. That is acceptable given the identical framework, but a referee may want more detail. The Zariski-density claim is inherited from the orbit structure in the standard way, though not spelled out.

Who gets value: analytic number theorists in the affine sieve program, and anyone tracking the constants in almost-prime results. It deserves a serious referee. I would send it back for revision to supply the joint equidistribution lemma and lay out the two-variable setup; if that lemma works out, this is close to a clean accept. If the joint error term turns out worse, the constants 16/14 may need adjustment.

Recommendation: send to peer review, require the missing two-variable argument before acceptance.

Referee Report

2 major / 4 minor

Summary. The paper proves two almost-prime results for integral points on a three-variable indefinite anisotropic quadratic form f(x)=t with td(f) square-free. Assuming V(Z) is non-empty, Theorem 1.1 states that the set of points with x1 having at most 6 prime factors outside the fixed bad set B={2,3,5,7} is Zariski dense, with 6 improved to 5 under Selberg's eigenvalue conjecture. Theorem 1.2 states the analogous Zariski-density result for the product x1x2 having at most 16 prime factors outside B, with 16 improved to 14 under Selberg's conjecture. The proofs combine an equidistribution estimate for the orbit (Lemma 2.3), Kim-Sarnak bounds via Jacquet-Langlands, and explicit one- and two-dimensional weighted sieves of Diamond-Halberstam and Halberstam-Richert.

Significance. If fully correct, the results are a genuine improvement over the earlier Liu-Sarnak theorem, replacing almost-primality of the product of all three coordinates with almost-primality of one or two coordinates. The numerical sieve computations are explicit, and the claimed constants 6 and 16 are not borrowed from prior work but arise from the stated level of distribution. The main deficiency is that Theorem 1.2 relies on an unstated and unproved two-variable equidistribution estimate, so the significance of the paper's second theorem depends on an external claim that the authors should be asked to justify.

major comments (2)

[§4.2 and §2.2] Theorem 1.2 sieves the product x1x2, but Lemma 2.3 is proved only for the sequence a_n(T) defined in (2.4) by x1=±n, so the congruence n≡0 mod d in (2.8) corresponds to d | x1. The two-dimensional sieve in §4.2 requires an analogue for a sequence such as b_n(T)=sum_{x1x2=±n} F_T(x), with a distribution estimate for d | x1x2. The sentence in §2 that 'the situation for two variables is similar' is not a proof, and no such lemma is stated or proved. This is load-bearing: the constants 16 and 14 in §4.2 are computed from (3.2) with τ=25/128 or τ=1/4, and those computations are only valid for a sequence whose equidistribution error has the same shape as (2.8). A joint estimate with error O(d^{2+ε} T^{1/2+θ+ε}) would change the admissible level of distribution and the numerical values of m(ζ). Please supply a complete statement and proof of the two-variable equidistribution, or an explicit derivation from [18].
[§4.2] The application of the nonlinear sieve in Lemma 3.2 is not fully specified: the paper does not define the finite sequence A used for Theorem 1.2, nor does it verify the two hypotheses max_{a_n∈A} n ≤ X^{τμ} and (3.2) for that sequence. For instance, the value μ=2/τ is asserted without stating what n represents for the product x1x2. This gap is partly formal, but it is connected to the missing joint equidistribution: one needs to know that the same level τ controls the error term for the product sequence before applying the sieve conclusion r>m(ζ).

minor comments (4)

[Title and Abstract] There are several conversion artifacts, such as 'V ariables' in the title and 's /greaterorequalslant3' in §1.1; these should be corrected during production.
[§2.2] The notation O^0(Z/dZ) is used both for the set with y1≡0 mod d and for the set with x1x2x3≡0 mod d; the two definitions should be distinguished to avoid confusion.
[§4.2] The displayed formula for m(ζ) contains an ambiguous term 'ζ 2 − µ β2'; adding parentheses, e.g. (2+ζ) log(β2/ζ) - 2 + (ζ^2 - μ)/β2, would clarify the algebra.
[§1.2] The phrase 'the above 6 can be reduced to 5' and the similar phrase for 16 and 14 are grammatically awkward; 'the constant 6 can be replaced by 5' would be clearer.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the constants 6 and 16 are computed outputs of the sieve framework, not inputs borrowed from [18].

full rationale

The paper's derivation chain is self-contained relative to the prior theorem [18], which is an independent published result (Liu–Sarnak) and not the target result of this paper. The main steps are: (i) define the weighted sequence a_n(T) from an orbit O; (ii) prove Lemmas 2.1–2.2 using estimates from [18]; (iii) state Lemma 2.3, whose proof is quoted as 'exactly the same as that of [18, Theorem 2.1]'; (iv) feed the resulting level of distribution τ = 1/4 − θ/2 = 25/128 into the weighted sieve (Proposition 3.5 for κ = 1 and Lemma 3.2 with the Halberstam–Richert estimate for κ = 2); (v) perform explicit numerical integrations to obtain r > 5.996 (unconditional) and r > 4.676 (conditional), giving r = 6 and 5, and similarly r > 15.6327 / 13.0287 giving r = 16 / 14. In every case, the claimed constants arise from these computations, not from fitting or from the theorem statement being assumed. The heavy citation of [18] provides the framework and technical equidistribution lemma, which is legitimate independent support: it is a previously proven theorem and does not depend on the current results. No equation is defined in terms of the target conclusion, and no fitted parameter is renamed as a prediction. One caveat is noted explicitly: Section 2 says 'the situation for two variables is similar' and Section 4.2 silently uses τ = 25/128 for the two-dimensional sieve, but no joint equidistribution lemma for (x1, x2) is stated or proved. This is a missing proof obligation and a correctness risk (if the two-variable error term had larger d-dependence, the computed constant 16 would fail), but it is not a circular reduction of the conclusion to the hypotheses.

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

The paper introduces no free parameters fitted to data. The sieve optimization choices (a=1, b=6.6, ζ) are analytic parameter choices, not fitted values. All substantive inputs are standard theorems or prior published work; the only assumed conjecture is Selberg's, used for conditional improvements.

assumptions (5)

standard math Local-global principle for anisotropic ternary quadratic forms (Siegel mass formula, one class in genus)
Introduction, before Theorem 1.1: ensures V(Z) is non-empty and splits into finitely many orbits once a single solution exists.
standard math Kim-Sarnak bound θ=7/64 towards Selberg eigenvalue conjecture (λ1 ≥ 1/4 - θ^2)
Section 2, (2.7): sets τ=25/128 in Lemma 2.3; used for the unconditional theorems.
standard math Jacquet-Langlands correspondence for automorphic forms on GL(2)
Section 2: transfers spectral bounds for SL2(R) to co-compact lattices from quaternion algebras.
domain assumption Orbit decomposition and weight functions from Liu-Sarnak [18]
Section 2: Lemma 2.3 proof and local density bounds refer to [18, Theorem 2.1 and (4.8)-(4.13)]; these are prior published results by one of the authors.
domain assumption Selberg eigenvalue conjecture (conditional statements only)
Theorems 1.1 and 1.2 second parts: assumed θ=0 giving τ=1/4 and improved constants 5 and 14.

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Pith. "Pith review of Anisotropic quadratic equations in three variables." pith.science (2026). https://pith.science/paper/U6D7N7JC

@misc{pith2026250115033,
  author       = {Pith},
  title        = {Pith review of: Anisotropic quadratic equations in three variables},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/U6D7N7JC}},
  note         = {Machine review of arXiv:2501.15033}
}

abstract

Let $f(x_1, x_2, x_3)$ be an indefinite anisotropic integral quadratic form with determinant $d(f)$, and $t$ a non-zero integer such that $d(f)t$ is square-free. It is proved in this paper that, as long as there is one integral solution to $f(x_1, x_2, x_3) = t$, there are infinitely many such solutions for which (i) $x_1$ has at most $6$ prime factors, and (ii) the product $x_1 x_2$ has at most $16$ prime factors. Various methods, such as algebraic theory of quadratic forms, harmonic analysis, Jacquet-Langlands theory, as well as combinatorics, interact here, and the above results come from applying the sharpest known bounds towards Selberg's eigenvalue conjecture. Assuming the latter the number $6$ or $16$ may be reduced to $5$ or $14$, respectively.

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Reference graph

Works this paper leans on

23 extracted references · 23 canonical work pages

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [18]

Liu and P

J. Liu and P. Sarnak, Integral points on quadrics in three varia bles whose coordinates have few prime factors, Israel J. Math. 178 (2010), 393–426

work page 2010

[2] [1]

Blomer and J

V. Blomer and J. Br¨ udern, A three squares theorem with almost primes, Bull. London Math. Soc. 37 (2005), 507–513

work page 2005

[3] [2]

Borovoi, On representations of integers by indeﬁnite ternar y quadratic forms, J

M. Borovoi, On representations of integers by indeﬁnite ternar y quadratic forms, J. Number Theory 90 (2003), 281–293

work page 2003

[4] [3]

Bourgain, A

J. Bourgain, A. Gamburd and P. Sarnak, Aﬃne linear sieve, expan ders, and sum-product, Invent. Math. 179 (2010), 559–644

work page 2010

[5] [4]

Bourgain, A

J. Bourgain, A. Kontorovich, The aﬃne sieve beyond expansion I : Thin hypotenuses, Int. Math. Res. Not. IMRN (2015), no. 19, 9175–9205

work page 2015

[6] [5]

J. W. S. Cassels, Rational quadratic forms, London Mathematical Society Monographs 13, Academic Press, London-New York, 1978

work page 1978

[7] [6]

Diamond, H

H. Diamond, H. Halberstam, Some applications of sieves of dimensio n exceeding 1, in: Sieve Meth- ods, Exponential sums, and their Applications in Number The ory (Cardiﬀ,1995) , London Mathe- matical Society Lecture Notes Series, Vol. 237, Cambridge Univers ity Press, Cambridge, 1997, pp. 101–107

work page 1995

[8] [7]

Proce dures for computing sieve functions

H. Diamond, H. Halberstam, A higher-dimensional sieve method , Cambridge Tracts in Math. no.177, Cambridge University Press, 2008; With an appendix “Proce dures for computing sieve functions” by William F. Galway

work page 2008