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Hardy-Littlewood varieties and semisimple groups. (English) Zbl 0917.11025
Let $$X(A)= X(\mathbb{R})\times X(A_f)$$ be the set of adelic points of an algebraic variety $$X$$ over $$\mathbb{Q}$$ (here $$A_f$$ is the $$\mathbb{Q}$$-algebra of finite adeles). Let $$B_f$$ be a compact subset of $$X(A_f)$$, let $$B_0(T)=\{x\mid x\in B_0$$, $$| x|\leq T\}$$ be the ball of radius $$T$$ in a connected component $$B_0$$ of $$X(\mathbb{R})$$, and let $$B(T)= B_0(T)\times B_f$$. The authors are interested in the asymptotic behaviour of the counting function ${\mathcal N}_X (T,B)= \text{card} \bigl(X(\mathbb{Q}) \cap B(T) \bigr),$ as $$T\to\infty$$. The variety $$X$$ is said to be strongly (respectively relatively) Hardy-Littlewood if $${\mathcal N}_X(T,B)\sim\mu(B(T))$$ (respectively $${\mathcal N}_X(T,B) \sim\int_{B(T)} \delta (x) d\mu(x)$$ for a suitable non-negative function $$\delta:X(A) \to \mathbb{R})$$, where $$\mu$$ is the Tamagawa measure on $$X(A)$$. The authors write: “We prove that certain affine homogeneous spaces are strongly Hardy-Littlewood, $$\dots$$ that many homogeneous spaces are relatively Hardy-Littlewood, but not strongly Hardy-Littlewood. This yields a new class of varieties, for which the asymptotic density of integer points can be computed in terms of a product of local densities”.
Reviewer: B.Z.Moroz (Bonn)

##### MSC:
 11G35 Varieties over global fields 14M17 Homogeneous spaces and generalizations 11N45 Asymptotic results on counting functions for algebraic and topological structures 11P55 Applications of the Hardy-Littlewood method
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