Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Proof of W.M.Schmidt's conjecture concerning successive minima of a
  lattice   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$.

11
The categories of this paper are: hep-lat
32
The categories of this paper are: atom-ph
24
The categories of this paper are: dg-ga
8
The categories of this paper are: hep-ex
5
The categories of this paper are: gr-qc
13
The categories of this paper are: quant-ph
4
The categories of this paper are: cond-mat
7
The categories of this paper are: hep-th
17
The categories of this paper are: q-fin
36
The categories of this paper are: cmp-lg
15
The categories of this paper are: nucl-ex
16
The categories of this paper are: stat
22
The categories of this paper are: chao-dyn
35
The categories of this paper are: mtrl-th
12
The categories of this paper are: nucl-th
0
The categories of this paper are: hep-ph