[Paper Review] Effective Ratner theorem for ASL(2,R) and gaps in \sqrt{n} modulo 1

T. D. Browning, Ilya Vinogradov|arXiv (Cornell University)|Nov 25, 2013

Algebraic Geometry and Number Theory10 references3 citations

TL;DR

This paper establishes an effective equidistribution rate for unipotent flows on the homogeneous space $\mathrm{SL}(2,\mathbb{R})\ltimes\mathbb{R}^2/\mathrm{SL}(2,\mathbb{Z})\ltimes\mathbb{Z}^2$, proving a $y^{1/4}\log^2(2+y^{-1})$ error bound for the equidistribution of horocycle lifts. The result resolves an open question by Elkies and McMullen on effective gap distribution for $\sqrt{n}\mod 1$, providing explicit error estimates for the limiting gap distribution.

ABSTRACT

Let G=ASL(2,R) be the affine special linear group of the plane, and set Gamma=ASL(2,Z). Building on recent work of Strömbergsson we prove a rate of equidistribution for the orbits of a certain 1-dimensional unipotent flow of Gamma\G, which projects to a closed horocycle in the unit tangent bundle to the modular surface. We use this to answer a question of Elkies and McMullen by making effective the convergence of the gap distribution of \sqrt{n} modulo 1.

Motivation & Objective

• To provide an effective rate of equidistribution for a 1-dimensional unipotent flow on $\Gamma\backslash G$, where $G = \mathrm{SL}(2,\mathbb{R})\ltimes\mathbb{R}^2$ and $\Gamma = \mathrm{SL}(2,\mathbb{Z})\ltimes\mathbb{Z}^2$, extending Strömbergsson's work.
• To resolve an open question by Elkies and McMullen concerning the effective convergence of the gap distribution of $\sqrt{n}\mod 1$.
• To establish quantitative bounds on the equidistribution of orbits under the $a(y)$-flow on the lift of a closed horocycle in the unit tangent bundle of the modular surface.
• To extend effective Ratner-type equidistribution results to a non-standard unipotent subgroup $U$ that is a codimension-1 submanifold of the full unstable manifold.
• To apply these bounds to the spectral gap problem for fractional parts of $\sqrt{n}$, yielding explicit error terms in the convergence to the limiting gap distribution.

Proposed method

• The authors use Fourier analysis and estimates for complete exponential sums, relying on Weil's bounds for character sums over finite fields.
• They apply a dyadic decomposition of the parameter space and exploit multiplicative properties of exponential sums to control error terms.
• The method involves interpolating between Sobolev norms $\|\rho\|_{W^{1,1}}$ and $\|\rho\|_{W^{2,1}}$ to handle compactly supported functions with limited smoothness.
• A key technical step is the use of bounds on exponential sums $T_{p^m}(A,B,C)$ via the theory of critical points and multiplicity in finite fields, particularly for $p \neq 3$ and $p = 3$.
• The proof leverages multiplicativity of exponential sums and bounds on divisor functions and $K^{\omega(n)}$ sums to control the growth of sums over $n$, $k$, and dyadic intervals.
• The authors combine estimates from different regimes (based on vanishing of $k$ or $l$) and apply $L^1$-based Sobolev norms to derive a final error bound with a $y^{1/4}$ decay rate.

Experimental results

Research questions

• RQ1 Can effective equidistribution rates be established for unipotent flows on $\mathrm{SL}(2,\mathbb{R})\ltimes\mathbb{R}^2/\mathrm{SL}(2,\mathbb{Z})\ltimes\mathbb{Z}^2$ when the unipotent subgroup is not the full unstable manifold?
• RQ2 What is the optimal rate of equidistribution for the orbit of $u(x) = \left(\begin{smallmatrix}1&x\\0&1\end{smallmatrix}\right), (x/2, x^2/4)$ under the $a(y)$-flow as $y \to 0$?
• RQ3 Can the ineffective equidistribution result of Elkies and McMullen for the gap distribution of $\sqrt{n}\mod 1$ be made effective with explicit error bounds?
• RQ4 What is the sharp decay rate in $y$ for the discrepancy between the average of a smooth function over the horocycle lift and its mean over $X$?
• RQ5 How do the smoothness properties of the test function and the weight $\rho$ affect the error rate in the equidistribution estimate?

Key findings

• The paper establishes an effective equidistribution rate of $O(y^{1/4}\log^2(2 + y^{-1}))$ for the average of a $C^8_b$ function over the horocycle lift, with the error measured against the invariant measure on $X$.
• For compactly supported $\rho$ with $1+\varepsilon$ derivatives in $L^1$, the error bound is $O\left(\|\rho\|_{W^{1,1}}^{1-\eta}\|\rho\|_{W^{2,1}}^{\eta}\|f\|_{C^8_b} y^{1/4}\log^{K-1}(2 + y^{-1})\right)$ for any $\eta \in (0,1)$, with an absolute constant $K$.
• The exponent $1/4$ in the error term is optimal for the method used, though the authors conjecture it could be improved to $1/2$ with stronger cancellation in two-dimensional exponential sums.
• The method successfully handles the case where the unipotent subgroup $U$ is a proper codimension-1 submanifold of the full unstable manifold, overcoming a key obstruction in prior approaches.
• The bounds are applied to the sequence $\sqrt{n}\mod 1$, yielding explicit error estimates for the convergence of the gap distribution to its limiting form.
• The result provides the first effective version of Elkies and McMullen’s equidistribution result for the gap distribution of $\sqrt{n}\mod 1$, answering their open question on explicit error terms.