Effective and explicit gap statistics of $\sqrt{un}\bmod 1$: the surviving open questions of Elkies and McMullen

Cut the unit circle at the fractional parts $\{\sqrt{1}\},\dots,\{\sqrt{N}\}$; the complementary intervals (gaps), normalized by the mean gap $1/N$, have a limiting distribution that is not exponential, by the theorem of Elkies and McMullen (Duke Math. J. 2004), proved via Ratner's theorem on the space of affine unimodular lattices. Their paper asked for effective and explicit versions of this result: error terms for the convergence, closed-form formulas for the truncated-window density $F^{(\theta)}(t)$ and for the dilated densities $F_u^{(\theta)}(t)$ when $u\in\mathbb{Q}$ and $2/u\notin\mathbb{Z}$, explicit joint distributions of consecutive gaps, and the conjectural exponential (Poisson) gap law for $\{\sqrt{un}\}$ with irrational $u$. Subsequent literature has made the convergence effective for the base sequence (Strömbergsson 2015; Browning and Vinogradov 2016; Vinogradov 2016) and produced a new circle-method proof (Radziwiłł and Technau 2024), while Poissonian pair correlation was proved by El-Baz, Marklof and Vinogradov (2015) and pigeonhole statistics by Pattison. The effective-rates question is now largely resolved, but a precise, nonempty open core survives: the explicit formulas, the structure of the multi-gap joint laws, and the irrational-dilation exponential law.

Background

For a real number $x$, write $\{x\}=x\bmod 1$ for its fractional part in $\mathbb{R}/\mathbb{Z}$. A classical theme of uniform distribution theory asks how evenly such sequences fill the circle; a sharper question is their fine-scale statistics, for which the lengths of the gaps between consecutive points are a standard probe. If $N$ points are placed independently and uniformly at random, the gaps $J$ normalized by the mean gap $1/N$ obey the exponential law $\#{J:|J|\in[a/N,b/N]}/N\to\int_a^b e^{-t}\,dt$, the gap law of a Poisson process; observing this law is regarded as evidence of pseudo-randomness.

Elkies and McMullen studied the gaps $J_1,\dots,J_N$ left when the circle is cut at $\{\sqrt{1}\},\dots,\{\sqrt{N}\}$. Their main theorem (Theorem 1.1) states that $N|J_i|$ is governed in the limit by a piecewise real-analytic density $F(t)$ with genuine phase transitions at $t=1/2$ and $t=2$: $F$ is constant (equal to $6/\pi^2$) on $[0,1/2]$ and has a power-law tail $F(t)\sim(3/\pi^2)t^{-3}$ rather than an exponential one. Thus $\{\sqrt{n}\}$ is conspicuously non-Poissonian at the level of gaps, in contrast to the conjectural exponential behavior of $\{n^{\beta}\}$ for $\beta\in(0,1)$, $\beta\neq 1/2$.

The proof connects the gap problem to homogeneous dynamics. A unimodular lattice in $\mathbb{R}^2$ is a discrete subgroup isomorphic to $\mathbb{Z}^2$ with quotient torus of area one; a lattice translate is a coset $v+\Lambda_0$. The space of all such translates is the homogeneous space $E=\mathrm{ASL}_2(\mathbb{R})/\mathrm{ASL}_2(\mathbb{Z})$ of dimension 5, where $\mathrm{ASL}_2$ denotes area-preserving affine maps; $E$ carries a unique invariant probability measure $\mu_E$. Elkies and McMullen show that the gap containing a random point $x\in[0,1]$ is controlled by a lattice translate $\Lambda_{s^2}(x)$ obtained by a linearization of $(a+x)^2$, and that the family $\Lambda_{s^2}(x)$ equidistributes in $E$ (their Theorem 1.2), a consequence of Ratner's theorem classifying measures invariant under unipotent flows (a unipotent flow is the action of a one-parameter subgroup of unipotent, i.e. parabolic, matrices such as $\begin{pmatrix}1&t\\0&1\end{pmatrix}$; the $\Lambda_{s^2}(x)$ traverse a nonlinear horocycle section, i.e. a curve whose projection to $\mathrm{SL}_2(\mathbb{R})/\mathrm{SL}_2(\mathbb{Z})$ is a closed horocycle, a circle of large period). Consequently, if $p(t)$ denotes the probability that a random lattice translate meets a fixed triangle of area $t$, then $p''(t)=-F(t)$; the paper evaluates this to an explicit elementary formula for $F$ (involving arctangents and logarithms, its Theorem 3.14).

Two generalizations frame the open questions. First, restricting the indices to the truncated window $\theta^2N<n\le N$ ($0<\theta<1$; the window then contains $(1-\theta^2)N+O(1)$ points, and the gaps are still measured in units of $1/N$, the number of gaps with length in $[t_0/N,t_1/N]$ being asymptotic to $N\int_{t_0}^{t_1}F^{(\theta)}(t)\,dt$) replaces the governing triangle by an isosceles trapezoid and yields a modified gap density $F^{(\theta)}(t)$, for which the paper obtains only a double-integral representation (its Theorem 3.15): $F^{(\theta)}$ is piecewise analytic, constant equal to $(1-\theta^2)^2/\zeta(2)$ on $[0,1/(2-2\theta)]$ (here $\zeta(2)=\sum_{n\ge1}n^{-2}=\pi^2/6$ is the Riemann zeta function at $2$), and has a $t^{-3}$ tail. Second, fixing a rational $u>0$ and real $\nu$, one may take the gaps of $\{\sqrt{un+\nu}\}$ with $0<un+\nu<N$; the gap law, with gaps measured in units of $u/N$ (the proportion of gaps with length in $[t_0u/N,t_1u/N]$ tends to $\int_{t_0}^{t_1}F_u(t)\,dt$), converges to a limit $F_u(t)$ independent of $\nu$ but depending on $u$, constructed from the periodic set $\omega_{1/u}\mathbb{Z}^2$, where $\omega_v(x,y)=(x,y+v^2x)$. Writing $u=u_1/u_2$ in lowest terms, $\omega_{1/u}\mathbb{Z}^2$ is a disjoint union of at most $u_1$ translates of a lattice and is itself a lattice translate only when $u_1\mid 2$; the corresponding moduli space is $E(u)=\mathrm{ASL}_2(\mathbb{R})/\Gamma(u)$, where $\Gamma(u)$ is the stabilizer of $\omega_{1/u}\mathbb{Z}^2$ in $\mathrm{ASL}_2(\mathbb{R})$, a congruence subgroup containing the $3\times3$ matrices in $\mathrm{ASL}_2(\mathbb{Z})$ congruent to the identity modulo $u_1$. Elementary closed forms are available only in the lattice case ($2/u\in\mathbb{Z}$); beyond the general existence theorem, Elkies and McMullen prove that $F_u(0)<1$ for all $u$, that $F_u(0)\to 1$ as $u_1\to\infty$, and give an orbit-sum formula for $F_u(0)$ when $u_1=p$ or $2p$ for an odd prime $p$. Finally, the joint distribution of $r$ consecutive normalized gaps converges by the same method to a law supported on a countable union of manifolds of dimension at most $5=\dim E$, whose part inside the simplex $\sum_j t_j<1/2$ has dimension exactly 2 (it comes from collinear lattice points and is therefore singular once $r>2$); with the truncated window the simplex becomes $\sum_j t_j<1/(2-2\theta)$.

Subsequent literature has developed the effective side of this theory: Strömbergsson (2015) and Browning and Vinogradov (2016) made the equidistribution of horocycle sections effective, the latter obtaining the quantitative gap-distribution error $|\lambda_N(t)-\lambda_\infty(t)|\ll_\varepsilon(1/t^2+t)N^{-1/68+\varepsilon}$, and Vinogradov (2016) treated general horocycle lifts; Strömbergsson, Södergren and Vishe (2026) generalized the effective equidistribution to $\mathrm{SL}_2(\mathbb{R})\ltimes(\mathbb{R}^2)^k$. Radziwiłł and Technau (2024) gave an essentially self-contained proof of the gap theorem by the circle method (a technique of analytic number theory that rewrites the relevant counting function as an integral over the unit circle, estimated separately on arcs near roots of unity and on the remaining arcs). Complementary fine-scale statistics were computed by El-Baz, Marklof and Vinogradov (2015), who proved that the two-point correlation function of $\{\sqrt{n}\}$ (the asymptotic density of pairs of points at a given small separation, scaled by the mean spacing) is Poissonian, and by Pattison (2023), who determined the pigeonhole statistics (the limit of the number of points of $\{\sqrt{n}\}$ falling in each of $N$ equal intervals). None of these addresses the explicit formulas for $F^{(\theta)}$ and $F_u$, the multi-gap joint laws, or the irrational-dilation regime.

Problem Statement

The common objective is to replace the qualitative existence theorems of Elkies and McMullen for the gap statistics of $\{\sqrt{un}\}$ by effective, explicit, and dilation-uniform descriptions. The problem is the surviving open core of their Section 3.6 'Open questions', with the rates component already substantially resolved by later work (see Background) and only its residue retained. A solution settles the following linked components; the first three constitute the core, and each asks for proofs, not numerical evidence alone.

(1) Explicit formulas. Determine whether the truncated-window gap density $F^{(\theta)}(t)$, which governs the normalized gaps of $\{\sqrt{n}\}$ restricted to $\theta^2N<n<N$, is expressible as an elementary function of $t$ and $\theta$, and if so, give the formula. Answer the same question for $F_u^{(\theta)}(t)$ for rational $u>0$ with $2/u\notin\mathbb{Z}$, the case not covered by the elementary formulas available when $2/u\in\mathbb{Z}$ (here $F_u^{(\theta)}$ is defined by the Elkies-McMullen construction via the periodic set $\omega_{1/u}\mathbb{Z}^2$ and the moduli space $E(u)$).

(2) Joint distributions. Determine whether the joint distribution of $r>1$ consecutive normalized gaps of $\{\sqrt{un}\}$ (including the $\theta$-truncated and rational-$u$ variants) can be obtained explicitly for any $r>1$; if not, at least locate the phase transitions and the supports of these joint distributions on $[0,\infty)^r$: identify which manifolds carry positive mass, and determine their dimensions, so as to sharpen the paper's bounds (support in manifolds of dimension at most 5; dimension exactly 2 inside the simplex $\sum_j t_j<1/2$, or $\sum_j t_j<1/(2-2\theta)$ for the truncated window).

(3) Irrational dilation. For irrational $u>0$: does $F_{u_i}(t)\to e^{-t}$ for each fixed $t\ge 0$ whenever $u_i$ are rationals approaching $u$ (the source paper notes this is known only for $u=0$)? Prove or refute the conjecture, suggested by numerical experiments, that the normalized gaps of $\{\sqrt{un}\}$ converge to the exponential (Poisson) gap law for fixed irrational $u$ - in full (over all $N\to\infty$) for some explicit class of irrational $u$, and also for almost all irrational $u$; and answer the same questions for the truncated-window densities $F_u^{(\theta)}$ and for the joint distributions of consecutive gaps.

(4) Residue of the rates question (secondary). Improve the effective convergence rate of the gap-counting functions beyond the known $N^{-1/68+\varepsilon}$ error, and determine which of the two error sources - the linear approximation of $(a+t)^2$ or the rate of equidistribution of the horocycle sections $\Lambda_{s^2}(x)$ - accounts for the observed finite-$N$ behavior at normalized gap sizes below $1/2$.

References

1. Noam D. Elkies and Curtis T. McMullen, 'Gaps in $\sqrt{n}$ mod 1 and ergodic theory', Duke Mathematical Journal 123 (2004), no. 1, 95-139, DOI 10.1215/s0012-7094-04-12314-0, https://doi.org/10.1215/s0012-7094-04-12314-0 (published version also openly available via the Harvard DASH persistent handle https://dash.harvard.edu/handle/1/3637161, direct bitstream https://dash.harvard.edu/bitstream/1/3637161/5/McMullen_GapsErgodoticTheory.pdf; Section 3.6, 'Open questions', of this paper poses the questions set out in the Problem Statement)
2. Noam D. Elkies and Curtis T. McMullen, 'Correction to Gaps in $\sqrt{n}$ mod 1 and ergodic theory', Duke Mathematical Journal 129 (2005), no. 2, DOI 10.1215/s0012-7094-05-12927-1, https://doi.org/10.1215/s0012-7094-05-12927-1
3. Andreas Strömbergsson, 'An effective Ratner equidistribution result for $\mathrm{SL}(2,\mathbb{R})\ltimes\mathbb{R}^2$', Duke Mathematical Journal 164 (2015), no. 5, 843-902, DOI 10.1215/00127094-2885873, https://doi.org/10.1215/00127094-2885873 (preprint arXiv:1309.6103, https://arxiv.org/abs/1309.6103, posted under the variant title 'An Effective Ratner Equidistribution Result for ASL(2,R)')
4. Tim Browning and Ilya Vinogradov, 'Effective Ratner theorem for $\mathrm{SL}(2,\mathbb{R})\ltimes\mathbb{R}^2$ and gaps in $\sqrt{n}$ modulo 1', Journal of the London Mathematical Society 94 (2016), no. 1, 61-84, DOI 10.1112/jlms/jdw025, https://doi.org/10.1112/jlms/jdw025 (preprint arXiv:1311.6387, https://arxiv.org/abs/1311.6387, v1 posted under the variant title 'Effective Ratner theorem for ASL(2,R) and gaps in $\sqrt{n}$ modulo 1'; later versions carry the published title)
5. Ilya Vinogradov, 'Effective equidistribution of horocycle lifts', arXiv:1607.04769, https://arxiv.org/abs/1607.04769 (preprint)
6. Maksym Radziwiłł and Niclas Technau, 'Gap distribution of $\sqrt{n}\,\mathrm{mod}\,1$ and the circle method', arXiv:2403.16493, https://arxiv.org/abs/2403.16493 (preprint)
7. Jens Marklof, 'Delone sets generated by square roots', American Mathematical Monthly 127 (2020), no. 9, 836-840, DOI 10.1080/00029890.2020.1811003, https://doi.org/10.1080/00029890.2020.1811003 (preprint arXiv:2003.08319, https://arxiv.org/abs/2003.08319)
8. Jens Marklof, 'Square roots and lattices', L'Enseignement Mathématique 72 (2025), no. 1, 175-190, DOI 10.4171/lem/1092, https://doi.org/10.4171/lem/1092 (preprint arXiv:2406.09107, https://arxiv.org/abs/2406.09107)
9. Daniel El-Baz, Jens Marklof, and Ilya Vinogradov, 'The two-point correlation function of the fractional parts of $\sqrt{n}$ is Poisson', Proceedings of the American Mathematical Society 143 (2015), no. 7, 2815-2828, DOI 10.1090/S0002-9939-2015-12489-6, https://doi.org/10.1090/S0002-9939-2015-12489-6 (preprint arXiv:1306.6543, https://arxiv.org/abs/1306.6543)
10. Sam Pattison, 'Rational points on nonlinear horocycles and pigeonhole statistics for the fractional parts of $\sqrt{n}$', Ergodic Theory and Dynamical Systems 43 (2023), no. 9, 3108-3130, DOI 10.1017/etds.2022.58, https://doi.org/10.1017/etds.2022.58 (preprint arXiv:2104.02569, https://arxiv.org/abs/2104.02569, posted under the hyphenated variant title 'Rational points on non-linear horocycles and pigeonhole statistics for the fractional parts of $\sqrt{n}$')
11. Christopher Lutsko, 'Gap statistics of the sequence $\{\alpha\sqrt{n}\}$', arXiv:2004.07646, https://arxiv.org/abs/2004.07646 (preprint; the arXiv listing records that a gap was found in the proof)
12. Christopher Lutsko, Athanasios Sourmelidis, and Niclas Technau, 'Pair correlation of the fractional parts of $\alpha n^{\theta}$', Journal of the European Mathematical Society 27 (2024), no. 10, 4069-4082, DOI 10.4171/JEMS/1449, https://doi.org/10.4171/JEMS/1449 (preprint arXiv:2106.09800, https://arxiv.org/abs/2106.09800)
13. Simon Čopar, 'Gaps in the fractional parts of square roots', arXiv:2012.14019, https://arxiv.org/abs/2012.14019 (preprint)
14. Andreas Strömbergsson, Anders Södergren, and Pankaj Vishe, 'Effective equidistribution of unipotent orbits in homogeneous spaces of $\mathrm{SL}(2,\mathbb{R})\ltimes(\mathbb{R}^2)^k$', arXiv:2604.08753, https://arxiv.org/abs/2604.08753 (preprint)