Effective density for inhomogeneous quadratic forms II: fixed forms and generic shifts

2020/01/29 by Anish Ghosh, Ghosh, Anish, Dubi Kelmer +3 · 2 citations
Mathematics · #Advanced Algebra and Geometry #Analytic Number Theory Research #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2001.10990

openalex publication_date 2020/01/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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).

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• Distribution of values at tuples of integer vectors under symplectic forms
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