Proof of W.M.Schmidt's conjecture concerning successive minima of a lattice

Abstract

For a real $N\ge 1$ and a vector $\xi =(1,{\xi }_1,...,{\xi }_n)$ define a matrix $$ {\cal A} (ξ, N) = ({array}{ccccc} N^{-1} & 0& 0& ... &0 \cr N^{\frac{1}{n}} ξ_1 & -N^{\frac{1}{n}} & 0&... & 0 \cr N^{\frac{1}{n}} ξ_2 &0& -N^{\frac{1}{n}} & ... & 0 \cr ... &... &... &... \cr N^{\frac{1}{n}} ξ_n &0&0&... &- N^{\frac{1}{n}} {array}) $$ and a lattice $$\Lambda (\xi ,N)=\mathcal{A}(\xi ,N){\mathbb{Z}}^{n+1}.$$ Consider a convex 0-symmetric body $$\mathcal{W}=\{z=(x,y_1,...,y_n)\in {\mathbb{R}}^{n+1}:\max (|x|,|y|)\le 1\}>.$$ For a natural $l,1\le l\le n+1$ let ${\mu }_l(\xi ,N)$ be the $l$-th successive minimum of $\mathcal{W}$ with respect to $\Lambda (\xi ,N)$. We prove that there exist real numbers ${\xi }_1,...,{\xi }_n$ linearly independent together with 1 over $\mathbb{Z}$, such that ${\mu }_k(\xi ,N)\to 0$ as $N\to \infty$ and ${\mu }_{k+2}(\xi ,N)\to \infty$ as $N\to \infty$.

Submitted to Proceedings of LMS, further minor corrections

References in corpus (1)

• The best Diophantine approximations: the phenomenon of degenerate dimension

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• A variational principle in the parametric geometry of numbers
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• Sets of inhomogeneous linear forms can be not isotropically winning