Integrability of Siegel transforms and an application

TL;DR

The paper develops a generalized Siegel transform $S_\chi$ for rational representations stabilizing a maximal parabolic of a semisimple $\mathbb{Q}$-group and identifies sharp $L^p$-integrability criteria for $p\in\{1,2,\infty\}$, connecting mean-value formulas, unimodularity, and lattice structure. It then proves an effective equidistribution for translated maximal-compact orbits under expanding diagonals, providing single and double equidistribution with explicit decay rates in Sobolev norms. These analytic results are applied to metric Diophantine approximation on rank-one flag varieties, yielding an effective Schmidt-type counting formula at the Diophantine exponent $\beta_{\chi}$: for a.e. $x\in X$, ${\mathcal N}_{c,βχ}(x,T) = κ\, c^{d} \log T\,(1+O_x(\log T^{-ε}))$, where $d=\dim X$. The approach combines invariant-measure formulas, Weil-type integration, and the effective equidistribution of translated $K$-orbits to obtain explicit asymptotics with error terms, thereby advancing understanding of Diophantine approximation on flag varieties via representation-theoretic integrability and dynamics.

Abstract

We establish sharp algebraic criteria for the $L^{p}$-integrability, for $p = 1, 2, \infty$, of a natural generalization of the Siegel transform to the setting of rational representations of semisimple algebraic $\mathbb{Q}$-groups, extending Siegel's analytic work in the geometry of numbers. As an application, we derive an effective asymptotic formula for the number of rational approximations of bounded height to almost every real point on a rank-one flag variety at the Diophantine exponent. The argument combines the integrability criterion with effective equidistribution estimates for translated orbits of maximal compact subgroups, a result of independent interest.

Figures (1)

• Figure 1: The set ${\mathcal{E}}_{\beta_{\chi}}(T)^+$ for the group $G = \mathop{\mathrm{SL}}\nolimits_2(\mathbb{R})$, the flag variety the real projective line $X = \mathbb{P}^1(\mathbb{R}) = \mathrm{Gr}_{1,2}(\mathbb{R})$, the punctured affine cone $\widetilde{X} = \mathbb{R}^2 \setminus \{0\}$ above $X$, and the set ${\mathcal{P}}_{\chi} = \mathcal{P}(\mathbb{Z}^2)$ of primitive elements of $\mathbb{Z}^2$. Rational approximations to a point $x \in X$ of height bounded by $T$ correspond to primitive lattice points of $\mathbb{Z}^2$ in the red region $k_x {\mathcal{E}}_{\beta_{\chi}}(T)$, where $k_x \in \mathop{\mathrm{SO}}\nolimits_2(\mathbb{R})$ is a rotation such that $x = k_x [\bm{e}_1]$. The action of $a(y) = \mathop{\mathrm{diag}}\nolimits ( y^{-1/2}, y^{1/2})$ with $y > 1$ on $\widetilde{X} = \mathbb{R}^2 \setminus \{0\}$ contracts the line through $\bm{e}_\chi = \bm{e}_1$ and expands the line through $\bm{e}_2$. The domain ${\mathcal{E}}_{\beta_{\chi}}(T)^+$ can be decomposed into translates of the elementary domain ${\mathcal{F}}$ under the action of $a(y)$. The hope is that for $x$ chosen randomly according to the Lebesgue measure on $X$ the number of primitive lattice points in the red region $k_x {\mathcal{E}}_{\beta_{\chi}}(T)$, that is, the quantity $\# ({\mathcal{P}}_{\chi} \cap k_x \, {\mathcal{E}}_{\beta_{\chi}}(T))$, which is the classical primitive Siegel transform of the indicator function $\mathbbm{1}_{{\mathcal{E}}_{\beta_{\chi}}(T)}$ evaluated at the rotated lattice $k_x^{-1} \mathbb{Z}^2$, is approximately given (up to a scalar) by the volume of ${\mathcal{E}}_{\beta_{\chi}}(T)$.

Theorems & Definitions (33)

• Definition 1.1: Siegel transform
• Theorem A: $L^1$-integrability
• Theorem B: $L^{\infty}$-integrability
• Theorem C: $L^2$-integrability
• Theorem D: Effective equidistribution of maximal compact subgroup orbits
• Theorem E: Effective counting at the Diophantine exponent
• Corollary 1.2
• Lemma 3.1
• proof
• Lemma 3.2