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 second moment of Siegel transforms on certain congruence quotients of $\operatorname{SL}_n(\mathbb{R})$ 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.

中文翻译：

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非齐次二次型的有效密度 I：通用型和固定位移

我们为通用非齐次二次型建立了奥本海姆猜想的有效版本。我们证明了固定移位向量和通用二次形式的这样的结果。当移动是合理的时，我们证明了一个计数结果，这意味着通用非均匀形式的值的最佳密度。对于满足显式丢番图条件的固定无理位移，我们还获得了类似的密度结果。主要的技术工具是一个公式，用于在 $\operatorname{SL}_n(\mathbb{R})$ 的某些同余商上进行 Siegel 变换的二阶矩，我们认为它具有独立意义。在续集中，我们使用不同的技术来处理关于泛型移位和固定二次形式的伴随问题。