[Paper Review] Effective density for inhomogeneous quadratic forms II: fixed forms and generic shifts

Anish Ghosh, Dubi Kelmer+1 more|arXiv (Cornell University)|Jan 29, 2020

Advanced Algebra and Geometry17 references4 citations

TL;DR

This paper establishes effective density results for inhomogeneous quadratic forms by proving that for fixed rational indefinite quadratic forms in $ n \geq 3 $ variables, and for almost all shifts $ \alpha \in \mathbb{R}^n $, the values $ Q_\alpha(v) $ approximate any real number $ \xi $ within $ t^{-\kappa} $ for $ \|v\| \leq t $, with explicit bounds on the exponent $ \kappa_0 $ depending on the signature of $ Q $. The method relies on effective mean ergodic theorems and spectral gap estimates for representations of orthogonal groups without Kazhdan's property (T).

ABSTRACT

We establish effective versions of Oppenheim's conjecture for generic inhomogeneous quadratic forms. We prove such results for fixed quadratic forms and generic shifts. Our results complement our companion paper where we considered generic forms and fixed shifts. In this paper, we use ergodic theorems and in particular we establish a strong spectral gap with effective bounds for some representations of orthogonal groups which do not possess Kazhdan's property (T).

Motivation & Objective

• To establish effective versions of Oppenheim's conjecture for inhomogeneous quadratic forms with fixed indefinite rational forms and generic shifts.
• To address the effectivity problem in homogeneous dynamics by quantifying how closely values of inhomogeneous quadratic forms can approximate real numbers.
• To derive explicit bounds on the approximation exponent $ \kappa_0 $ depending on the signature of the quadratic form.
• To extend the understanding of shrinking target problems in the context of group actions on homogeneous spaces.
• To develop effective spectral gap estimates for representations of orthogonal groups lacking Kazhdan’s property (T), enabling effective ergodic theorems.

Proposed method

• Reduces the problem to a shrinking target problem for the action of the group $ G = \mathrm{SO}^+_Q(\mathbb{R}) $ on a homogeneous space.
• Applies an effective mean ergodic theorem to control the decay rate of averages over orbits, linking it to spectral properties of unitary representations.
• Uses induction on the dimension $ n $, reducing the problem to lower-rank subgroups $ G^{(j)} \cong \mathrm{SO}^+(n-j,1) $, with recursive spectral gap bounds.
• Employs the parameter $ \alpha(\pi) = \frac{n-2}{p(\pi)} $ to characterize matrix coefficient decay, enabling recursive control of the spectral gap.
• Analyzes unitary representations $ \tilde{\pi} = \mathrm{Ind}_{\tilde{G}^\lambda}^{\tilde{G}} \sigma $ of the semidirect product $ \tilde{G} = G \ltimes \mathbb{R}^n $, focusing on those with no $ \mathbb{R}^n $-invariant vectors.
• Applies a reduction argument from Burger and Sarnak to relate the spectral gap of $ \pi|_{G^{(j)}} $ to that of the full representation, enabling inductive control of $ p(\pi) \leq \kappa(n-2) $.

Experimental results

Research questions

• RQ1 For a fixed rational indefinite quadratic form $ Q $, what is the optimal exponent $ \kappa_0 $ such that $ |Q_\alpha(v) - \xi| < t^{-\kappa} $ has integer solutions for all large $ t $, for almost all shifts $ \alpha $?
• RQ2 How does the approximation rate depend on the signature $ (p,q) $ of the quadratic form?
• RQ3 Can effective spectral gap estimates be established for representations of $ \mathrm{SO}^+(p,q) $ that do not possess Kazhdan’s property (T)?
• RQ4 To what extent can effective mean ergodic theorems be applied to shrinking target problems in the absence of spectral gap from property (T)?
• RQ5 What is the relationship between the spectral gap of a representation and the decay rate of matrix coefficients restricted to a Cartan subgroup?

Key findings

• For any rational indefinite quadratic form $ Q $ in $ n \geq 3 $ variables, and for almost all shifts $ \alpha \in \mathbb{R}^n $, the system $ |Q_\alpha(v) - \xi| < t^{-\kappa} $, $ \|v\| \leq t $, has integer solutions for all sufficiently large $ t $, provided $ \kappa < \kappa_0 $.
• The exponent $ \kappa_0 $ is explicitly computed based on the signature $ (p,q) $: for example, $ \kappa_0 = 1 $ when $ (p,q) = (2,1) $, $ \kappa_0 = 2 $ for $ (n-1,1) $, and $ \kappa_0 = 3/2 $ for $ (4,2) $ or $ (3,3) $.
• For general signatures with $ p \geq q > 1 $, $ \kappa_0 = 2\kappa_1 q(p-1) $, where $ \kappa_1 $ depends on $ n \mod 4 $, giving $ \kappa_0 \approx n/2 $ when $ p \approx q $.
• The method yields the optimal bound $ \kappa_0 = n-2 $ for $ n = 3,4 $, but for $ n \geq 5 $, the bound is weaker than the expected $ n-2 $, especially for $ (n-1,1) $ where $ \kappa_0 = 2 $.
• The proof establishes a strong effective spectral gap for representations of $ \mathrm{SO}^+(p,q) $ without Kazhdan’s property (T), using induction and recursive control of matrix coefficient decay.
• The result confirms that effective mean ergodic theorems can be applied to non-(T) groups via a reduction to lower-rank subgroups, extending techniques from [ GGN20 ] and [ GK17 ].