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

TL;DR

This paper proves a conjecture by W.M. Schmidt regarding the behavior of successive minima of a specific lattice constructed from a vector and parameter N, showing certain minima tend to zero while others tend to infinity.

Contribution

It establishes the existence of vectors with linearly independent components for which the successive minima exhibit prescribed asymptotic behavior as N grows.

Findings

Existence of vectors with prescribed minima behavior

Successive minima can be controlled independently

Asymptotic divergence and convergence of minima

Abstract

For a real $N\ge 1$ and a vector $\xi =(1,\xi_1,...,\xi_n)$ define a matrix $$ {\cal A} (\xi, N) = ({array}{ccccc} N^{-1} & 0& 0& ... &0 \cr N^{\frac{1}{n}} \xi_1 & -N^{\frac{1}{n}} & 0&... & 0 \cr N^{\frac{1}{n}} \xi_2 &0& -N^{\frac{1}{n}} & ... & 0 \cr ... &... &... &... \cr N^{\frac{1}{n}} \xi_n &0&0&... &- N^{\frac{1}{n}} {array}) $$ and a lattice $$ \Lambda (\xi, N) = {\cal A} (\xi, N)\mathbb{Z}^{n+1}. $$ 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 $\mu_l (\xi, N)$ be the $l$-th successive minimum of ${\cal W}$ 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$.