Birkhoff genericity on affine subspaces in horospheres

Technical Summary: Birkhoff Genericity on Affine Subspaces in Horospheres

Problem Statement
The paper investigates the Birkhoff genericity of points on affine subspaces within the expanding horospherical orbit of a simple uniformly expanding diagonal flow on the space of unimodular lattices, $X = \mathrm{SL}_{n+1}(\mathbb{R})/\mathrm{SL}_{n+1}(\mathbb{Z})$. Specifically, the authors consider the action of a one-parameter diagonal subgroup $\{a_t\}$ on points $u x_0$, where $u$ varies over a proper affine subspace $A$ of the expanding horospherical subgroup $U^+_a \cong \mathbb{R}^n$.

While the Birkhoff ergodic theorem guarantees that almost every point in the entire space $X$ is generic, and previous results established genericity for translates of smooth measures on submanifolds under diagonal actions, this work addresses the specific question of whether the initial points restricted to an affine subspace $A$ are generic. The authors seek to determine the conditions under which the time averages of continuous functions along the diagonal orbit of points in $A$ converge to the space average with respect to the $G$-invariant probability measure $\mu_X$.

Methodology
The proof strategy extends techniques developed by Shi, Shah, and others, integrating dynamical systems methods with Diophantine approximation. The methodology proceeds through several key stages:

1. Non-escape of Mass: The authors first establish that for almost every point in the affine subspace, the trajectory does not "escape to infinity" (i.e., the mass of the time-averaged measures does not concentrate at the cusp). This is achieved by constructing a Margulis height function $\alpha_\varepsilon$ and analyzing its exponential growth rate along trajectories of a commuting diagonal subgroup $\{b_t\}$. They relate this growth rate to the Diophantine exponent $\omega(A)$ of the defining matrix of the affine subspace. If $\omega(A) < n$, the growth rate is controlled, ensuring non-divergence.

2. Unipotent Invariance and Measure Rigidity: Using the non-escape result and Ratner's measure rigidity theorems for unipotent flows, the authors show that any weak-$*$ limit of the time-averaged measures is invariant under a unipotent subgroup. If such a limit measure is not the full $G$-invariant measure, it must be supported on a "singular submanifold" (a closed orbit of a proper algebraic subgroup).

3. Avoidance of Singular Submanifolds: To prove that the limit measure is indeed the full $G$-invariant measure, the authors must show that the trajectories avoid these singular submanifolds. They construct a specialized height function $\beta_N$ relative to the singular submanifold $N$. This function is designed to contract under the action of the flow $\{a_t\}$ (specifically via the $c_t$ component of the flow) while expanding under the $\{b_t$ component.

4. Diophantine Obstructions: The core of the argument links the failure of the height function to contract (which would imply the trajectory stays near the singular submanifold) to specific Diophantine properties of the affine subspace. The authors demonstrate that if the trajectory does not equidistribute, the affine subspace must satisfy one of two specific "bad" Diophantine conditions:

   • The defining matrix has a Diophantine exponent $\omega(A) \ge n$.
   • The affine subspace is "Liouville" in a specific sense: it is arbitrarily well-approximated by lower-dimensional affine subspaces defined over a real number field $F$ of degree $m \ge 2$, where $n+1 = mr$.

Key Contributions and Results

• Main Theorem (Theorem 1.2): The authors prove that if the Diophantine exponent $\omega(A) < n$, then for almost every point $u \in A$, the orbit is Birkhoff generic, unless the affine subspace is $(F, r-1)$-Liouville for some real number field $F$ of degree $m \ge 2$ satisfying $n+1 = mr$.

  • A significant corollary is that if $n+1$ is prime, possibility (2) cannot occur; thus, for prime $n+1$ and $\omega(A) < n$, almost every point is Birkhoff generic.
  • The condition $\omega(A) < n$ is shown to be optimal; if $\omega(A) > n$, mass escape occurs for every point in the subspace.

• Dirichlet Non-Improvability (Theorem 1.4): As an application, the authors deduce that for almost every point on such affine subspaces, Dirichlet's approximation theorem cannot be improved. Specifically, the sets of Dirichlet-improvable vectors intersected with the affine subspace have Lebesgue measure zero within the subspace. This result weakens the conditions required in previous work (specifically for the case $d=1, n \ge 3$) to establish non-improvability.

• Logarithmic Density Results (Theorem 1.5): The paper establishes precise logarithmic density results for the set of times $T$ for which Dirichlet's approximation can be improved by a factor $\delta$. For almost every point on the subspace, these densities are given by continuous, strictly increasing functions $f(\delta)$ and $g(\delta)$, independent of the specific point.

• Novel Height Function Construction: A technical novelty of the paper is the construction of a height function $\beta_N$ specifically tailored to measure the distance from singular submanifolds associated with number fields. This function combines the standard Margulis height with a geometric term measuring the "distance" to the singular set, satisfying specific contraction properties under the diagonal flow.

Significance and Claims
The paper claims to provide a complete characterization of Birkhoff genericity for affine subspaces in horospheres, identifying the precise Diophantine obstructions that prevent genericity. The authors position their work as a complement to existing results on non-degenerate curves and subgroup actions (citing works by Shi, Kleinbock, Weiss, Solan, and Wieser).

The significance lies in:

1. Generalization: Extending Birkhoff genericity results from curves and general submanifolds to the specific and structured case of affine subspaces.
2. Optimality: Demonstrating that the Diophantine exponent condition is sharp; the paper explicitly notes that if the exponent is too large, genericity fails entirely due to mass escape.
3. Diophantine Applications: Providing a dynamical proof for the non-improvability of Dirichlet's theorem on these subspaces and quantifying the frequency of improvements via logarithmic density.
4. Liouville Phenomena: Highlighting a specific "Liouville" obstruction related to number fields of degree $m \ge 2$ that prevents equidistribution, a phenomenon that does not occur when $n+1$ is prime.

The authors do not claim to resolve the critical case where $\omega(A) = n$, nor do they propose new experimental applications, maintaining a focus on the theoretical classification of genericity and its Diophantine consequences.