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

Abstract

For a real $N\ge 1$ and a vector $ξ=(1,ξ_1,...,ξ_n)$ 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 $$ Λ(ξ, N) = {\cal A} (ξ, N)\mathbb{Z}^{n+1}. $$ Consider a convex 0-symmetric body $${\cal W} = \{z= (x,y_1,...,y_n)\in \mathbb{R}^{n+1}: \max (|x|, |y|)\le 1 \} >.$$ For a natural $l, 1\le l \le n+1$ let $μ_l (ξ, N)$ be the $l$-th successive minimum of ${\cal W}$ with respect to $ Λ(ξ, N)$. We prove that there exist real numbers $ξ_1,...,ξ_n$ linearly independent together with 1 over $\mathbb{Z}$, such that $μ_k (ξ, N) \to 0$ as $ N\to \infty$ and $μ_{k+2} (ξ, N) \to \infty$ as $ N\to \infty$.