Moments of Margulis functions and indefinite ternary quadratic forms
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911064878219264 |
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| author | Kim, Wooyeon |
| author_facet | Kim, Wooyeon |
| contents | In this paper, we prove a quantitative version of the Oppenheim conjecture for indefinite ternary quadratic forms: for any indefinite irrational ternary quadratic form $Q$ that is not extremely well approxiable by rational forms, and for $a<b$ the number of integral vectors of norm at most $T$ satisfying $a<Q(v)<b$ is asymptotically equivalent to $\big(\mathsf{C}_Q(b-a)+\mathsf{I}_{Q}(a,b)\big)T$ as $T$ tends to infinity, where the constant $\mathsf{C}_Q>0$ depends only on $Q$, and the term $\mathsf{I}_{Q}(a,b)T$ accounts for the contribution from rational isotropic lines and degenerate planes.
The main technical ingredient is a uniform bound for the $λ$-moment of the Margulis $α$-function along expanding translates of a unipotent orbit in $\operatorname{SL}_3(\mathbb{R})/\operatorname{SL}_3(\mathbb{Z})$, for some $λ>1$. To establish this, we introduce a new height function $\widetildeα$ on the space of lattices, which captures the failure of the classical Margulis inequality. This moment bound implies equidistribution of such translates with respect to a class of unbounded test functions, including the Siegel transform. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_16563 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Moments of Margulis functions and indefinite ternary quadratic forms Kim, Wooyeon Dynamical Systems Number Theory In this paper, we prove a quantitative version of the Oppenheim conjecture for indefinite ternary quadratic forms: for any indefinite irrational ternary quadratic form $Q$ that is not extremely well approxiable by rational forms, and for $a<b$ the number of integral vectors of norm at most $T$ satisfying $a<Q(v)<b$ is asymptotically equivalent to $\big(\mathsf{C}_Q(b-a)+\mathsf{I}_{Q}(a,b)\big)T$ as $T$ tends to infinity, where the constant $\mathsf{C}_Q>0$ depends only on $Q$, and the term $\mathsf{I}_{Q}(a,b)T$ accounts for the contribution from rational isotropic lines and degenerate planes. The main technical ingredient is a uniform bound for the $λ$-moment of the Margulis $α$-function along expanding translates of a unipotent orbit in $\operatorname{SL}_3(\mathbb{R})/\operatorname{SL}_3(\mathbb{Z})$, for some $λ>1$. To establish this, we introduce a new height function $\widetildeα$ on the space of lattices, which captures the failure of the classical Margulis inequality. This moment bound implies equidistribution of such translates with respect to a class of unbounded test functions, including the Siegel transform. |
| title | Moments of Margulis functions and indefinite ternary quadratic forms |
| topic | Dynamical Systems Number Theory |
| url | https://arxiv.org/abs/2403.16563 |