Hardy-Littlewood varieties and semisimple groups
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''.
asymptotic behaviour of the counting function
scientific article; zbMATH DE number 763281
scientific article