Moments of Margulis functions and indefinite ternary quadratic forms

2024/03/25 by Kim, Wooyeon · 1 citation
#Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2403.16563

Abstract

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>0 depends only on Q, and the term 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 SL₃(ℝ)/SL₃(ℤ), for some λ&gt;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.</b>

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