Primes in thin orbits, Selberg-quality spectral gaps for thin congruence subgroups, and effective saturation numbers: the unresolved core of the Bourgain–Gamburd–Sarnak 'Sieving and expanders' program

The 2006 note 'Sieving and expanders' by Bourgain, Gamburd and Sarnak initiated the affine sieve: a combinatorial (Brun) sieve for orbits of finitely generated subgroups of GL_n(Z) whose Zariski closure is large (e.g. isomorphic to SL_2), powered by expansion of congruence graphs and by spectral gaps for thin congruence subgroups. Its almost-prime conclusions are now theorems in essentially maximal generality: Golsefidy–Varjú (2012) established congruence-graph expansion exactly for perfect Zariski closures, and Golsefidy–Sarnak (2013) proved saturation for all Levi-semisimple closures, strictly containing the semisimple connected simply connected case the note explicitly asked about. The open core of that program consists of three components: (i) the primary component, the quantitative non-Abelian Schinzel conjecture — genuine prime values of polynomials on thin orbits, with a Hardy–Littlewood-type asymptotic and a local-density constant; (ii) a spectral gap of arithmetic (Selberg 1/4) quality for thin congruence subgroups, where only 3/16-level existence results are known, most recently extended to geometrically finite thin subgroups of SO(n,1) in 2026; and (iii) effective, near-optimal saturation numbers, where current algorithms terminate but yield astronomically large values. These three components are, respectively, the sieve target, its spectral input, and its quantitative output within one program.

Background

Let $\Lambda$ be a finitely generated subgroup of $\mathrm{GL}_n(\mathbb{Z})$ acting on $\mathbb{Z}^n$, let $V=\Lambda b$ be the orbit of an integer point $b$, and let $f_1,\dots,f_t$ be integer polynomials. One asks whether the $f_j$ can take prime, or almost prime, values simultaneously at points of $V$, and how many such points there are. The abelian case — $V$ contained in a translate of a lattice in $\mathbb{Z}^k$ — is the setting of Dirichlet's theorem on primes in arithmetic progressions, of the Hardy–Littlewood prime $k$-tuple conjectures, and of Schinzel's hypothesis H. By the classical combinatorial (Brun) sieve — a method for counting integers or lattice points that avoid prime divisors below a given bound — such statements hold when 'prime' is weakened to '$r$-almost prime' (a product of at most $r$ primes), with $r$ depending on $k$; genuine primes are blocked by the parity problem, the structural inability of a sieve to distinguish numbers having an odd from an even number of prime factors.

The note (Bourgain–Gamburd–Sarnak, 'Sieving and expanders', C. R. Math. Acad. Sci. Paris 343 (2006) 155–159) launched the non-Abelian (affine) sieve. A group $\Lambda$ is called thin if its Zariski closure $\operatorname{Zcl}(\Lambda)$ — the smallest algebraic subgroup of $\mathrm{GL}_n$ over $\mathbb{Q}$ containing $\Lambda$ — is large (e.g. isomorphic to $\mathrm{SL}_2$) while $\Lambda$ has infinite index in the integral points of its closure; its orbit is then a sparse, non-linear analogue of an arithmetic progression. For $\operatorname{Zcl}(\Lambda)=\mathrm{SL}_2$ the note proves: (Theorem 1) there is an $r$, depending on $\Lambda$, $b$ and the $f_j$, such that the points of $V$ at which every $f_j$ is an $r$-almost prime are Zariski dense in the closure of $V$, with an 'effective, though very poor' dependence of $r$; (Theorem 2) for $\Lambda\subset\mathrm{SL}_2(\mathbb{Z})$ whose limit set (the set of accumulation points, on the boundary of the hyperbolic plane $\mathbb{H}$, of any orbit of $\Lambda$) has Hausdorff dimension $\delta(\Lambda)>1/2$, a sharp upper bound for the number of orbit points, ordered by Archimedean norm, at which the $f_j$ are all prime; (Theorem 3) the congruence graphs — graphs whose vertices are the reductions $V(q)$ of the orbit modulo $q$, with edges given by the generators of $\Lambda$ — form an expander family (a family of sparse graphs with optimal mixing) for square-free $q$, building on Bourgain–Gamburd's expansion bounds for Cayley graphs of $\mathrm{SL}_2(\mathbb{F}_p)$ (a Cayley graph has the elements of the group as vertices, with two elements joined when they differ by one of the fixed generators) and on Helfgott's product-growth theorem in $\mathrm{SL}_2(\mathbb{Z}/p\mathbb{Z})$ (small subsets of $\mathrm{SL}_2(\mathbb{Z}/p\mathbb{Z})$ grow rapidly under set multiplication); (Theorem 4) a sum–product estimate in $\mathbb{Z}/q\mathbb{Z}$ for square-free $q$ (any subset of $\mathbb{Z}/q\mathbb{Z}$ that is not close to a subring has a much larger sumset or productset); and (Theorem 5) a spectral gap: on the hyperbolic surface $X_\Lambda=\Lambda\backslash\mathbb{H}$ (of infinite volume when $\Lambda$ is thin), whose Laplacian (the Laplace–Beltrami operator of the hyperbolic metric, the analogue of the Euclidean Laplace operator) has finitely many eigenvalues $0\le\lambda_0<\lambda_1\le\cdots<1/4$ below $1/4$, one has $\lambda_1(\Lambda(q))\ge\lambda_0(\Lambda(q))+\varepsilon(\Lambda)$ for the congruence subgroups $\Lambda(q)=\{x\in\Lambda:x\equiv I\bmod q\}$ and all square-free $q$ — an infinite-volume analogue of Selberg's theorem that congruence subgroups of $\mathrm{SL}_2(\mathbb{Z})$ have no eigenvalues below $3/16$. Selberg's eigenvalue conjecture itself — the $1/4$ threshold — is the natural quality target for Theorem 5. The note states the expectation that Theorem 1 holds whenever $\operatorname{Zcl}(\Lambda)$ is semisimple, connected and simply connected, the only missing input being the combinatorics of Theorem 3 beyond $\mathrm{SL}_2$; it records that the quantitative non-Abelian Schinzel conjecture — the counting limit in Theorem 2 exists and is nonzero — is conjectural; and it defers full proofs to a longer paper ('Spectral sieving of thin sets', which never appeared under that title).

The subsequent literature carried this program out. Bourgain–Gamburd–Sarnak, 'Affine linear sieve, expanders, and sum-product' (Invent. Math. 179 (2010) 559–644) formalized saturation: a pair (orbit $O$, polynomial $f$) is said to saturate if, for some finite set $S'$ of primes and some finite $r$, the points of $O$ at which $f$ has at most $r$ prime factors outside $S'$ are still Zariski dense in the closure of $O$; the minimal such $r$ is the saturation number. Product-growth and expansion were extended beyond $\mathrm{SL}_2$, culminating in Golsefidy–Varjú, 'Expansion in perfect groups' (Geom. Funct. Anal. 22 (2012) 1832–1891): the congruence Cayley graphs of a finitely generated $\Gamma$ (for square-free moduli with large prime factors) form an expander family exactly when the connected component of $\operatorname{Zcl}(\Gamma)$ is perfect — equal to its own commutator subgroup, equivalently semisimple up to a unipotent radical. Bourgain–Varjú (Invent. Math. 188 (2012) 151–173) removed the square-free restriction on the modulus at the group level, and Longo–Golsefidy (2022) treated part of the positive-characteristic case. On top of this, Golsefidy–Sarnak, 'The affine sieve' (J. Amer. Math. Soc. 26 (2013) 1085–1105) proved the saturation conjecture for every Levi-semisimple Zariski closure (one whose quotient by its unipotent radical, the largest connected normal unipotent subgroup, is semisimple) — a class strictly larger than the semisimple connected simply connected groups the 2006 note asked about — so the almost-prime Theorem 1 is now available in essentially maximal generality; the boundary is sharp, since toral closures fail to saturate in general (Kontorovich–Lagarias). In parallel, Bourgain–Gamburd–Sarnak, 'Generalization of Selberg's 3/16 theorem and affine sieve' (Acta Math. 207 (2011) 255–290) proved Theorem 5, established the sharp upper bound (its Theorem 1.6) and the almost-prime lower bound (Theorem 1.7) for the Archimedean-ordered counting problem on thin subgroups of $\mathrm{SL}_2(\mathbb{Z})$, and stated as its conjecture (1.3) the prime asymptotic that is the heart of the present record. Nevo–Sarnak (Acta Math. 205 (2010) 361–402) executed the sieve with genuine primes in the 'algebraic' case of lattice orbits, using bounds toward the Ramanujan conjecture.

What remains open from the 2006 note is therefore precise: genuine primes on thin orbits, with their conjectural density; the quality of the spectral gap toward Selberg's $1/4$ for thin congruence subgroups (for thin $\Lambda$ the gap $\lambda_1(\Lambda(q))\ge\lambda_0(\Lambda(q))+\varepsilon(\Lambda)$ is known only with an ineffective $\varepsilon(\Lambda)$: no explicit numerical value is available for general $\delta(\Lambda)>1/2$ — explicit thresholds exist only under Gamburd's assumption $\delta(\Lambda)>5/6$ — and the quality of the known gaps matches Selberg's $3/16$ theorem for congruence subgroups of $\mathrm{SL}_2(\mathbb{Z})$, not the conjectured $1/4$; only recently has even existence been extended to geometrically finite thin subgroups of $\mathrm{SO}(n,1)$); and effective, near-optimal saturation numbers (terminating algorithms exist, but give 'absurdly large' values, and the best explicit values in special thin families are $R=4$ or $5$).

Problem Statement

The problem is the surviving open core of the source note, with one primary component and two explicitly linked auxiliary components.

Primary component (the quantitative non-Abelian Schinzel conjecture; conjecture (1.3) of Bourgain–Gamburd–Sarnak, Acta Math. 207 (2011), arXiv:0912.5021). Let $\Lambda\subset\mathrm{SL}_2(\mathbb{Z})$ be a finitely generated subgroup that is Zariski dense in $\mathrm{SL}_2$ (thin, i.e. of infinite index, in the cases of interest), and let $f\in\mathbb{Z}[x_{11},x_{12},x_{21},x_{22}]$ be weakly primitive on the orbit $O=\Lambda$ (the greatest common divisor of its values on $O$ is $1$), with factorization $f=f_1\cdots f_t$ into $t=t(f)$ irreducible factors in the coordinate ring $\mathbb{Q}[\mathrm{SL}_2]=\mathbb{Q}[x_{ij}]/(x_{11}x_{22}-x_{12}x_{21}-1)$. Let $\Vert\cdot\Vert$ be the Frobenius norm, $N_\Lambda(T)=\#\{\gamma\in\Lambda:\Vert\gamma\Vert\le T\}$ (known to grow like $c_\Lambda T^{2\delta(\Lambda)}$ when $\delta(\Lambda)>1/2$), and $\pi_{\Lambda,f}(T)=\#\{\gamma\in\Lambda:\Vert\gamma\Vert\le T,\ f_j(\gamma)\ \text{prime for every}\ j\}$. Assuming there is no local congruence obstruction to the $f_j$ being prime on $O$ — that is, no prime $p$ such that for every $x\in O$ at least one $f_j(x)$ is divisible by $p$ — prove the asymptotic

$$\pi_{\Lambda,f}(T)\ \sim\ c(\Lambda,f)\,\frac{N_\Lambda(T)}{(\log T)^{t(f)}}\qquad(T\to\infty),$$

where $c(\Lambda,f)$ is a nonzero product of local densities (an obstruction of the above kind forces the corresponding limit to be zero, and the conjecture is that this is the only way it can vanish). The qualitative statement that prime values of $f$ on $O$ are Zariski dense in the orbit closure whenever no local obstruction exists is weaker than, and implied by, this asymptotic. The same question is open for the word-length ordering (ordering $\Lambda$ by the length of its elements as words in a fixed finite generating set) and for orbits $V=\Lambda b\subset\mathbb{Z}^n$ of subgroups $\Lambda\subset\mathrm{GL}_n(\mathbb{Z})$ whose Zariski closure is any semisimple connected simply connected group, with several polynomials required prime simultaneously; these settings are optional variants of the primary question rather than further parts of it — establishing the primary question only in one of them answers that variant, not the primary component. The almost-prime versions of all of these are already theorems (Golsefidy–Sarnak 2013), so the new content is exclusively the passage from almost primes to primes and the density constant.

Auxiliary component (P2): spectral gap of arithmetic quality for thin congruence subgroups. Improve Theorem 5 of the source note ($\lambda_1(\Lambda(q))\ge\lambda_0(\Lambda(q))+\varepsilon(\Lambda)$ for $\delta(\Lambda)>1/2$ and all square-free $q$) in the direction of Selberg's $1/4$ threshold: prove an explicit uniform bound of arithmetic quality — for example $\lambda_1(\Lambda(q))\ge 1/4-c$ with an absolute explicit $c<1/16$ (beating Selberg's $3/16=1/4-1/16$ bound for congruence subgroups of $\mathrm{SL}_2(\mathbb{Z})$), or an analogue of the best arithmetic-case bounds, the Kim–Sarnak bound $\lambda\ge 1/4-(7/64)^2$ for congruence subgroups of $\mathrm{SL}_2(\mathbb{Z})$ — for all thin $\Lambda\subset\mathrm{SL}_2(\mathbb{Z})$ and all square-free $q$. Making $\varepsilon(\Lambda)$ explicit for general $\delta(\Lambda)>1/2$ is also open (explicit numerical values are currently available only under Gamburd's assumption $\delta(\Lambda)>5/6$).

Auxiliary component (P3): effective and near-optimal saturation numbers. For a given orbit data set $(\Lambda,b,f_1,\dots,f_t)$, determine $r$ effectively, with good (practically useful) dependence on $\Lambda$, $b$ and the $f_j$ — in particular, determine or approximate the saturation number, the minimal $r$ for which there is a finite set $S'$ of primes such that the points of the orbit at which the product $f_1\cdots f_t$ has at most $r$ prime factors outside $S'$ are Zariski dense — improving on the terminating-but-'absurdly large' algorithms currently available and on the best explicit values ($R=4$, $R=5$) known in special thin families. An explicit improvement on these benchmarks is also open: a smaller explicit $r$, for a stated family of orbit data, together with the finite set $S'$ of excluded primes for which the conclusion holds.

Relationship among the components: the exponent (level) of distribution available in the affine sieve is bounded by the spectral gap, which links (P2) to the primary component and to smaller $r$ and is why current levels of distribution and current $r$ are far from what the conjectures predict; (P3) is the quantitative refinement, on the same orbits, of the already-proven almost-prime theorem. The primary component implies the problem posed here; (P2) and (P3) are independent alternatives.

References

1. J. Bourgain, A. Gamburd, P. Sarnak, 'Sieving and expanders', C. R. Math. Acad. Sci. Paris 343 (2006), no. 3, 155–159. DOI: 10.1016/j.crma.2006.05.023. https://comptes-rendus.academie-sciences.fr/mathematique/item/10.1016/j.crma.2006.05.023/
2. J. Bourgain, A. Gamburd, P. Sarnak, 'Affine linear sieve, expanders, and sum-product', Inventiones Mathematicae 179 (2010), no. 3, 559–644. DOI: 10.1007/s00222-009-0225-3. https://doi.org/10.1007/s00222-009-0225-3
3. J. Bourgain, A. Gamburd, P. Sarnak, 'Generalization of Selberg's 3/16 theorem and affine sieve', Acta Mathematica 207 (2011), no. 2, 255–290. arXiv:0912.5021. DOI: 10.1007/s11511-012-0070-x. https://arxiv.org/abs/0912.5021
4. A. Salehi Golsefidy, P. Sarnak, 'The affine sieve', Journal of the American Mathematical Society 26 (2013), no. 4, 1085–1105. arXiv:1109.6432. DOI: 10.1090/S0894-0347-2013-00764-X. https://arxiv.org/abs/1109.6432
5. A. Salehi Golsefidy, P. P. Varjú, 'Expansion in perfect groups', Geometric and Functional Analysis 22 (2012), no. 6, 1832–1891. arXiv:1108.4900. DOI: 10.1007/s00039-012-0190-7. https://arxiv.org/abs/1108.4900
6. J. Bourgain, P. Varjú, 'Expansion in SL_d(Z/qZ), q arbitrary', Inventiones Mathematicae 188 (2012), no. 1, 151–173. DOI: 10.1007/s00222-011-0345-4. https://doi.org/10.1007/s00222-011-0345-4
7. J. Bourgain, A. Gamburd, 'Uniform expansion bounds for Cayley graphs of SL_2(F_p)', Annals of Mathematics (2) 167 (2008), no. 2, 625–642. DOI: 10.4007/annals.2008.167.625. https://doi.org/10.4007/annals.2008.167.625
8. H. A. Helfgott, 'Growth and generation in SL_2(Z/pZ)', Annals of Mathematics (2) 167 (2008), no. 2, 601–623. DOI: 10.4007/annals.2008.167.601. https://doi.org/10.4007/annals.2008.167.601
9. A. Gamburd, 'On the spectral gap for infinite index "congruence" subgroups of SL_2(Z)', Israel Journal of Mathematics 127 (2002), no. 1, 157–200. DOI: 10.1007/BF02784530. https://doi.org/10.1007/BF02784530
10. A. Nevo, P. Sarnak, 'Prime and almost prime integral points on principal homogeneous spaces', Acta Mathematica 205 (2010), no. 2, 361–402. arXiv:0902.0692. DOI: 10.1007/s11511-010-0057-4. https://arxiv.org/abs/0902.0692
11. A. Kontorovich, 'From Apollonius to Zaremba: local-global phenomena in thin orbits', arXiv:1208.5460 (2012). https://arxiv.org/abs/1208.5460
12. P. Sarnak, 'Notes on thin matrix groups', arXiv:1212.3525 (2012). https://arxiv.org/abs/1212.3525
13. A. Kontorovich, 'Levels of distribution and the affine sieve', arXiv:1406.1375 (2014). https://arxiv.org/abs/1406.1375
14. J. Bourgain, A. Kontorovich, 'The affine sieve beyond expansion I: thin hypotenuses', International Mathematics Research Notices 2015 (2015), no. 19, 9175–9205. arXiv:1307.3535. DOI: 10.1093/imrn/rnu222. https://arxiv.org/abs/1307.3535
15. J. Hong, A. Kontorovich, 'Almost prime coordinates for anisotropic and thin Pythagorean orbits', Israel Journal of Mathematics 209 (2015), no. 1, 397–420. arXiv:1401.4701. DOI: 10.1007/s11856-015-1223-3. https://arxiv.org/abs/1401.4701
16. M. Magee, 'Quantitative spectral gap for thin groups of hyperbolic isometries', Journal of the European Mathematical Society 17 (2015), no. 1, 151–187. arXiv:1112.2004. DOI: 10.4171/JEMS/500. https://arxiv.org/abs/1112.2004
17. A. Mohammadi, H. Oh, 'Matrix coefficients, counting and primes for orbits of geometrically finite groups', Journal of the European Mathematical Society 17 (2015), no. 4, 837–897. arXiv:1208.4139. DOI: 10.4171/JEMS/520. https://arxiv.org/abs/1208.4139
18. J. Bourgain, A. Kontorovich, 'On the local-global conjecture for integral Apollonian gaskets', Inventiones Mathematicae 196 (2014), no. 3, 589–650. DOI: 10.1007/s00222-013-0475-y. https://doi.org/10.1007/s00222-013-0475-y
19. T. Horesh, A. Nevo, 'Prime points in orbits: some instances of the Bourgain–Gamburd–Sarnak conjecture', Journal of Number Theory 193 (2018), 1–36. arXiv:1707.04746. DOI: 10.1016/j.jnt.2018.04.013. https://arxiv.org/abs/1707.04746
20. P. Sarkar, 'Generalization of Selberg's 3/16 theorem for geometrically finite thin subgroups of SO(n,1)', arXiv:2606.18674 (2026). https://arxiv.org/abs/2606.18674
21. A. Kontorovich, J. Lagarias, 'On toric orbits in the affine sieve', Experimental Mathematics 30 (2021), no. 4, 575–586. arXiv:1808.03235. DOI: 10.1080/10586458.2019.1616337. https://doi.org/10.1080/10586458.2019.1616337
22. B. Longo, A. Salehi Golsefidy, 'Toward super-approximation in positive characteristic', Journal of the London Mathematical Society 105 (2022), no. 2, 1200–1261. arXiv:1908.07014. DOI: 10.1112/jlms.12535. https://doi.org/10.1112/jlms.12535