Abstract
Let $G=$SL$(2,R)\ltimes(R^2)^{\oplus k}$ and let $\Gamma$ be a congruence subgroup of SL$(2,Z)\ltimes(Z^2)^{\oplus k}$. We prove a polynomially effective asymptotic equidistribution result for special types of unipotent orbits in $\Gamma\backslash G$ which project to pieces of closed horocycles in SL$(2,Z)\backslash$SL$(2,R)$. As an application, we prove an effective quantitative Oppenheim type result for the quadratic form $(m_1-\alpha)^2+(m_2-\beta)^2-(m_3-\alpha)^2-(m_4-\beta)^2$, for $(\alpha,\beta)$ of Diophantine type, following the approach by Marklof [24] using theta sums.
Highlights
The results of Ratner on measure rigidity and equidistribution of orbits of a unipotent flow [31, 32], play a fundamental role in homogeneous dynamics
In [3, 38], effective equidistribution results were obtained for orbits of a 1-parameter unipotent flow on SL(2, Z) Z2\ SL(2, R) R2, using Fourier analysis and methods of from analytic number theory, and in the very recent paper [30], building on similar methods, effective equidistribution of diagonal translates of certain orbits in SL(3, Z) Z3\ SL(3, R) R3 was established
Our purpose in the present paper is to prove results of a similar nature for homogeneous spaces of the group G = SL(2, R) (R2)⊕k for k 2, and to apply these to derive an effective quantitative Oppenheim-type result for a certain family of inhomogeneous quadratic forms of signature (2,2)
Summary
The results of Ratner on measure rigidity and equidistribution of orbits of a unipotent flow [31, 32], play a fundamental role in homogeneous dynamics. Let us note that Theorem 1.4 implies an effective version of the main theorem of [24], which says that under explicit Diophantine conditions on (α, β) ∈ R2, the local twopoint correlations of the sequence given by the values of Q1(m, n) = (m − α)2 + (n − β), with (m, n) ∈ Z2, are those of a Poisson process — a result which partly confirms a conjecture of Berry and Tabor [1] on quantized integrable systems. There exists an absolute constant B > 0 such that for any [κ; c]Diophantine vector (α, β) ∈ R2, and any real numbers a < b and Λ 1, This corollary gives an effective version of Marklof [24, Theorem 1.8], as well as of [23, Theorem 1.6] in the case k = 2, since the right-hand side of (16) tends to zero as T → ∞ for any fixed κ-Diophantine vector (α, β) (any κ) such that 1, α, β are linearly independent over Q. It is clear from this that if one would solve the aforementioned problem of proving an effective version of Theorem 1.1 in the general case with both ξ1, ξ2 allowed to be non-zero, this can be expected to lead to an effective version of [23, Theorem 5.1], and so, with further work, should lead to an effective version of [23, Theorem 1.6] for general k 2
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