当前位置: X-MOL 学术Monatshefte Math. › 论文详情
Our official English website, www.x-mol.net, welcomes your feedback! (Note: you will need to create a separate account there.)
Khintchine-type theorems for values of subhomogeneous functions at integer points
Monatshefte für Mathematik ( IF 0.9 ) Pub Date : 2021-01-03 , DOI: 10.1007/s00605-020-01498-1
Dmitry Kleinbock , Mishel Skenderi

This work has been motivated by recent papers that quantify the density of values of generic quadratic forms and other polynomials at integer points, in particular ones that use Rogers’ second moment estimates. In this paper, we establish such results in a very general framework. Given any subhomogeneous function (a notion to be defined) $$f: \mathbb {R}^n \rightarrow \mathbb {R}$$ f : R n → R , we derive a necessary and sufficient condition on the approximating function $$\psi $$ ψ for guaranteeing that a generic element $$f\circ g$$ f ∘ g in the G -orbit of f is $$\psi $$ ψ -approximable; that is, $$|f\circ g(\mathbf {v})| \le \psi (\Vert \mathbf {v}\Vert )$$ | f ∘ g ( v ) | ≤ ψ ( ‖ v ‖ ) for infinitely many $$\mathbf {v}\in \mathbb {Z}^n.$$ v ∈ Z n . We also deduce a sufficient condition in the case of uniform approximation. Here G can be any closed subgroup of $$\mathrm {ASL}_n(\mathbb {R})$$ ASL n ( R ) satisfying certain axioms that allow for the use of Rogers-type estimates.

中文翻译:

整数点上次齐次函数值的 Khintchine 型定理

这项工作受到最近的论文的推动,这些论文量化了整数点处的通用二次型和其他多项式的值的密度,特别是使用罗杰斯二阶矩估计的那些。在本文中,我们在一个非常通用的框架中建立了这样的结果。给定任何次齐次函数(要定义的概念)$$f: \mathbb {R}^n \rightarrow \mathbb {R}$$ f : R n → R ,我们推导出近似函数 $ $\psi $$ ψ 用于保证 f 的 G 轨道中的通用元素 $$f\circ g$$ f ∘ g 是 $$\psi $$ ψ -approximable;即 $$|f\circ g(\mathbf {v})| \le \psi (\Vert \mathbf {v}\Vert )$$ | f ∘ g ( v ) | ≤ ψ ( ‖ v ‖ ) 对于无穷多个 $$\mathbf {v}\in \mathbb {Z}^n.$$ v ∈ Z n 。我们还推导出了均匀近似情况下的充分条件。
更新日期:2021-01-03
down
wechat
bug