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On Schmidt and Summerer parametric geometry of numbers. (English) Zbl 1328.11076
W. M. Schmidt and L. Summerer [Acta Arith. 140, No. 1, 67–91 (2009; Zbl 1236.11060); Monatsh. Math. 169, No. 1, 51–104 (2013; Zbl 1264.11056)] provided new theory concerning the successive minima of certain convex bodies and they called it the parametric geometry of numbers, and they showed how this theory can be used to recover certain important inequalities relating standard exponents of Diophantine approximation attached to points of $$\mathbb{R}^n$$, and to find new ones.
In the present paper the main goal of the author is to simplify and to complete some aspects of this new theory. The author considers rigid systems, and proves that if $$n\geq 2$$ and $$\delta\in(0,\infty)$$, then for each unite vector $$u\in\mathbb{R}^n$$, there is a rigid system $$\mathbb{P}$$ with mesh $$\delta$$ such that $$\mathbb{L}_u-\mathbb{P}$$ is bounded on $$[q_0,\infty)$$; and conversely, for every rigid system $$\mathbb{P}$$ with mesh $$\delta$$, there is a unite vector $$u\in\mathbb{R}^n$$, such that $$\mathbb{L}_u-\mathbb{P}$$ is bounded on $$[q_0,\infty)$$. Here $$\mathbb{L}_u= (L_{u,1},\dots, L_{u,n})$$, and $$L_{u,j}$$ is the $$j$$th successive minima of convex bodies $$\{x\in\mathbb{R}^n+\| x\|\leq 1,|x\cdot u|\leq e^{-q}\}(q\geq 0)$$.

MSC:
 11H06 Lattices and convex bodies (number-theoretic aspects) 11J13 Simultaneous homogeneous approximation, linear forms 11J82 Measures of irrationality and of transcendence
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References:
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