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We study the light-cone Siegel transform, transforming functions on the light cone of a rational indefinite quadratic form Q to a function on the homogenous space $$\SO_Q(\Z)\backslash \SO_Q(\R)$$. In particular, we prove a second moment formula for this transform for forms of signature (n+1,1), and show how it can be used for various applications involving counting integer points on the light cone. In particular, we prove some new results on intrinsic Diophantine approximations on ellipsoids as well as on the distribution of values of random linear and quadratic forms on the light cone.
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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 previous paper [13] 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 $(T)$.more » « less
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Abstract In this work, we give an explicit formula for the Fourier coefficients of Eisenstein series corresponding to certain arithmetic lattices acting on hyperbolic ‐space. As a consequence, we obtain results on location of all poles of these Eisenstein series as well as their supremum norms. We use this information to get new results on counting rational points on spheres.more » « less
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Abstract We establish effective versions of Oppenheim’s conjecture for generic inhomogeneous quadratic forms. We prove such results for fixed shift vectors and generic quadratic forms. When the shift is rational we prove a counting result, which implies the optimal density for values of generic inhomogeneous forms. We also obtain a similar density result for fixed irrational shifts satisfying an explicit Diophantine condition. The main technical tool is a formula for the 2nd moment of Siegel transforms on certain congruence quotients of $$SL_n(\mathbb{R}),$$ which we believe to be of independent interest. In a sequel, we use different techniques to treat the companion problem concerning generic shifts and fixed quadratic forms.more » « less
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