Improving the level of distribution of thin Pythagorean hypotenuse sets beyond 7/24 (Bourgain–Kontorovich, Beyond Expansion I)

For a geometrically finite, Zariski-dense subgroup $\Gamma$ of the integer points of the Pythagorean group $\mathrm{SO}(x^2+y^2-z^2)$ with no nontrivial unipotent elements and critical exponent $\delta$ sufficiently close to 1, the set $S$ of hypotenuses of the Pythagorean triples in a thin orbit satisfies $\#(S\cap[-N,N])\sim C N^\delta$. Bourgain and Kontorovich proved, by a dispersion method that bypasses the spectral-gap barrier of the classical affine sieve, that $S$ admits an exponent of distribution $\alpha=7/24-\varepsilon$ for any $\varepsilon>0$ whenever $\delta>\delta_0(\varepsilon)$, yielding infinitely many hypotenuses with at most $R=4$ prime factors. The paper leaves open whether the exponent can be pushed past $7/24$ — concretely to $7/24+\varepsilon_0$ (Remarks 1.9 and 5.18) — how far toward an Elliott–Halberstam-type range the level of distribution of $S$ can go (Remark 1.19), and whether the requirement that $\delta$ be extremely close to 1 can be relaxed. No improvement of the exponent $7/24$ is known in the literature to date.

Background

A Pythagorean triple is an integral point on the cone $F(x,y,z)=x^2+y^2-z^2=0$, and its hypotenuse is the coordinate $z$. The connected component of the identity, $G=\mathrm{SO}^\circ(2,1)$, of the special orthogonal group of $F$ preserves integrality on this cone, so for a fixed triple $x_0$ (say $(1,0,1)$) and a subgroup $\Gamma<G(\mathbb{Z})$ one can study the orbit $\mathcal{O}=x_0\cdot\Gamma$ and the set $S$ of hypotenuses $f(x,y,z)=z$ of triples in $\mathcal{O}$. The orbit is called thin when $\Gamma$ has infinite index in the $\mathbb{Z}$-points of its Zariski closure — the smallest algebraic variety (zero set of polynomial equations in the matrix entries) containing it; $\Gamma$ is Zariski-dense when that closure is the whole ambient group. Of interest here are finitely generated, geometrically finite (admitting a fundamental region in the hyperbolic plane with finitely many sides), Zariski-dense such $\Gamma$ with no unipotent elements besides the identity, a unipotent element being a matrix all of whose eigenvalues equal 1 — in this two-dimensional setting, a parabolic element fixing a single point on the circle at infinity.

The group carries a critical exponent $\delta=\delta_\Gamma\in(0,1]$: the abscissa of convergence of its Poincaré series, $\sum_{\gamma\in\Gamma}e^{-s\,d(i,\gamma i)}$ with $d$ the hyperbolic distance on the upper half-plane and $i$ the base point; equivalently, it is the Hausdorff dimension of the limit set, the set of accumulation points of any orbit of $\Gamma$ on the boundary circle. Such a group is a lattice — a discrete subgroup of finite hyperbolic covolume — exactly when $\delta=1$, and the orbit is thinner the smaller $\delta$ is. For $\delta>1/2$ one has the counting law $\#(S\cap[-N,N])\sim C N^\delta$ (Kontorovich; Kontorovich–Oh), so a thin orbit ($\delta<1$) produces a genuinely sparse set of hypotenuses. The affine sieve of Bourgain, Gamburd, and Sarnak extends the classical sieve to such orbits, studying the prime factorization of polynomial values on them. One weights the points of $\mathcal{O}$ by nonnegative coefficients supported on $S\cap[-N,N]$, computes for each modulus $q$ a main term governed by local densities $\beta(q)$ (here of linear-sieve type, $\beta(p)\approx 1/p$ on average) and a remainder $r(q)$, and declares a level of distribution $Q$ when the total error $\sum_{q<Q}|r(q)|$ is smaller than the total mass by a power saving; writing $Q=N^\alpha$, the number $\alpha$ is the exponent of distribution. Larger $\alpha$ feeds the weighted sieve — the classical combinatorial sieve augmented by multiplicative weights attached to the prime factorization of the surviving elements, which sharpens the almost-prime output — and produces points whose $f$-value has fewer prime factors (an $R$-almost prime is an integer with at most $R$ prime factors; the least such $R$ for a sequence is its saturation number).

The standard route to a level of distribution is via the spectral gap of $\Gamma$ acting on congruence covers, the quotients of the hyperbolic plane by the subgroups obtained by reducing the matrix entries of $\Gamma$ modulo $q$; the gap is conventionally encoded by a parameter $\theta$, with $\theta$ closer to $1/2$ meaning a stronger gap. Gamburd's gap $\theta=5/6$ for infinite-index subgroups of $\mathrm{SL}_2(\mathbb{Z})$ gives $\alpha<1/12$ and hence $R=13$ for thin hypotenuses (Kontorovich; Kontorovich–Oh), and even a best-possible gap $\theta=1/2$ would cap this route at $\alpha<1/4$, giving $R=5$. Because the spin double cover — the two-to-one covering homomorphism $\mathrm{SL}_2(\mathbb{R})\to\mathrm{SO}^\circ(2,1)$ — is quadratic, a ball of size $N$ in the orbit corresponds to matrices of size $T$ with $N\sim T^2$, so $\alpha=1/4$ corresponds to level $Q\sim T^{1/2}$. Bourgain and Kontorovich call $1/4$ the natural analogue for this problem of the Bombieri–Vinogradov range (for primes in arithmetic progressions, the level $x^{1/2}$ of uniform distribution over progressions), and any exponent exceeding $1/4$ an instance of the Elliott–Halberstam range (the conjectural level $x^{1-\varepsilon}$ for primes in progressions). Bourgain and Kontorovich broke the spectral-gap barrier unconditionally for thin hypotenuses. Developing bilinear forms and the classical dispersion method (Cauchy–Schwarz bilinearization of incomplete exponential sums) in this thin-group setting, they proved (Theorem 1.6) that for every $\varepsilon>0$ there is $\delta_0(\varepsilon)<1$ such that, whenever $\delta>\delta_0(\varepsilon)$, the hypotenuse sequence admits exponent of distribution $\alpha=7/24-\varepsilon$; the weighted sieve then yields infinitely many hypotenuses in $S$ with at most $R=4$ prime factors (Corollary 1.8). The value $7/24$ results from a delicate balance of interlocking ranges in the dispersion argument: the modulus sum is split at a height $Q_0$, and five bilinear sums (denoted $S_1,\dots,S_5$) are estimated; the $S_5$ sum is disposed of trivially via $|S_5|\le|S_2|$, and the authors note (Remarks 1.9 and 5.18) that exploiting the further cancellation in $S_5$ (extra savings of $\tilde{q}^{\eta}$ and $(q_1q_1')^{\eta}$) 'may get the level just a hair higher', i.e. to $\alpha=7/24+\varepsilon_0$ for a tiny $\varepsilon_0>0$, an improvement for which they state no application is known. They also state that the threshold values $\delta_0(\varepsilon)=1-10^{-15}\varepsilon$ and $\delta_0=1-10^{-17}$ produced by the proof 'can surely be improved' (Remark 1.11), and that the method requires $\delta$ extremely close to 1, within the natural baseline $\delta>1/2$ of the counting and spectral machinery.

Subsequent work has clarified the landscape but not moved the exponent. Hong and Kontorovich discovered that sieving along a larger stabilizer subgroup doubles the exponent of distribution obtainable from expansion alone, improving the almost-prime records for areas and products of coordinates of Pythagorean triples; Ehrman adapted the Bourgain–Kontorovich dispersion machine to those functions and improved the admissible threshold for hypotenuses from $1-10^{-17}$ to $\delta_0=0.984$, while explicitly not improving the hypotenuse exponent or the value $R=4$; the companion papers Beyond Expansion II–IV applied the same machinery to low-lying and reciprocal geodesics on the modular surface (the quotient of the hyperbolic plane by $\mathrm{SL}_2(\mathbb{Z})$) and to traces of thin semigroups. Recent explicit spectral-gap results for Schottky-type subgroups — finitely generated free groups of hyperbolic isometries (Calderón–Magee; Soares) — strengthen only the expansion route, which, as the source notes, cannot raise the beyond-expansion exponent at all. As of the 2025 literature the exponent $7/24$ for thin hypotenuses still stands (Li–Liu still quote $R=4$ as the record), and the ultimate goal of producing primes in thin hypotenuse sets remains out of reach.

Problem Statement

Let $F(x,y,z)=x^2+y^2-z^2$, and let $\Gamma<\mathrm{SO}^\circ_F(\mathbb{Z})$ — the group of integer points in the identity component of the special orthogonal group of $F$ — be finitely generated, geometrically finite (admitting a fundamental region with finitely many sides), and Zariski-dense, with no unipotent elements other than the identity (a unipotent being a matrix all of whose eigenvalues equal 1) and with critical exponent $\delta\in(1/2,1)$ (the abscissa of convergence of the Poincaré series $\sum_{\gamma\in\Gamma}e^{-s\,d(i,\gamma i)}$, equivalently the Hausdorff dimension of the limit set of $\Gamma$). Let $S$ be the set of hypotenuses $z$ of the Pythagorean triples in the orbit $\mathcal{O}=x_0\cdot\Gamma$ — via the spin double cover (the two-to-one covering homomorphism $\mathrm{SL}_2(\mathbb{R})\to\mathrm{SO}^\circ_F(\mathbb{R})$), the values $c^2+d^2$ attained by the bottom rows $(c,d)$ of matrices in the lift of $\Gamma$ to $\mathrm{SL}_2(\mathbb{Z})$ — so that $\#(S\cap[-N,N])\sim C N^\delta$. Say that $S$ admits exponent of distribution $\alpha$ if there exist nonnegative sieve weights supported on $S\cap[-N,N]$, of total mass of order $N^\delta$, whose distribution among residue classes modulo $q$ has the expected main term with local densities $\beta(q)$ satisfying $\prod_{w\le p<z}(1-\beta(p))^{-1}\le C\log z/\log w$ and linear-sieve type $\beta(p)\approx 1/p$, and whose total remainder $\sum_{q<Q}|r(q)|$ is smaller than the total mass by a fixed power saving at level $Q=N^\alpha$ — exactly the framework of Theorem 1.6 of Bourgain–Kontorovich. Bourgain and Kontorovich proved that for every $\varepsilon>0$ there is $\delta_0(\varepsilon)<1$ such that $S$ admits exponent of distribution $\alpha=7/24-\varepsilon$ whenever $\delta>\delta_0(\varepsilon)$, and consequently that $S$ contains infinitely many $R=4$ almost primes (integers with at most four prime factors). The problem is to improve the unconditional exponent of distribution of this same sequence; resolving any one of the following linked facets of that single objective, at the strength stated for that facet, resolves the problem. (a) First milestone, from Remarks 1.9 and 5.18 of the source: prove that for some explicit $\varepsilon_0>0$ there is $\delta_0(\varepsilon_0)<1$ such that $S$ admits the exponent $\alpha=7/24+\varepsilon_0$ whenever $\delta>\delta_0(\varepsilon_0)$. (b) Elliott–Halberstam-type range, from Remark 1.19: determine how far past the 'Bombieri–Vinogradov' threshold $\alpha=1/4$ the level of distribution of $S$ can be pushed and what the true (conjectural) limit for this thin sequence is; a rigorous exponent $\alpha>7/24$, or an established limitation on the achievable level that is strictly stronger than the source's own bounds and that identifies precisely which estimate fails and where, resolves this facet. (c) Range of thinness: the known results require $\delta$ extremely close to 1 (the source's $\delta_0=1-10^{-17}$; the best published value $\delta_0=0.984$, due to Ehrman); extend beyond-expansion exponents (any $\alpha>1/4$) to an explicit threshold $\delta_0\le 0.9$, strictly below the best published value, ideally the full natural range $\delta>1/2$ of the counting framework. An answer must concern the same objects — the hypotenuse set $S$ of a thin Pythagorean orbit and its affine-sieve level of distribution — and must not replace them by a proxy (a different orbit or function, the area or product functions, or the lattice case $\delta=1$). Producing primes in $S$, or an asymptotic local–global principle for thin hypotenuses (the statement that $S$ captures, up to the expected density, the integers that are locally represented, i.e. represented modulo every modulus), would subsume this problem but is not required for it.

References

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