Vojta’s refinement of the subspace theorem
Authors: Schmidt Wolfgang M. 
Published: June 2012

Vojta’s refinement of the Subspace Theorem says that given linearly independent linear forms

L
1

,
…
,

L
n

{L_1}, \ldots , {L_n}

in

n
n

variables with algebraic coefficients, there is a finite union

U
U

of proper subspaces of

Q

n

{\mathbb {Q}^n}

, such that for any

ε
>
0

\varepsilon > 0

the points

x
_

_

∈

Z

n

∖
{

0
_

_

}

\underline {\underline x} \in {\mathbb {Z}^n}\backslash \{ \underline {\underline 0} \}

with (1)
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue upper L 1 left-parenthesis x underbar underbar right-parenthesis midline-horizontal-ellipsis upper L Subscript n Baseline left-parenthesis x underbar underbar right-parenthesis EndAbsoluteValue greater-than StartAbsoluteValue x underbar underbar EndAbsoluteValue Superscript negative epsilon">

|

L
1

(

x
_

_

)
⋯

L
n

(

x
_

_

)

|

>

|

x
_

_

|

−
ε

|{L_1}(\underline {\underline x} ) \cdots {L_n}(\underline {\underline x} )|\; > \;|\underline {\underline x} {|^{ - \varepsilon }}

lie in

U
U

, with finitely many exceptions which will depend on

ε
\varepsilon

. Put differently, if

X
(
ε
)

X(\varepsilon )

is the set of solutions of (1), if

X
¯

(
ε
)

\bar X(\varepsilon )

is its closure in the subspace topology (whose closed sets are finite unions of subspaces) and if

X
¯

′
(
ε
)

\bar X\prime (\varepsilon )

consists of components of dimension

>
1

> 1

, then

X
¯

′
(
ε
)
⊂
U

\bar X\prime (\varepsilon ) \subset U

. In the present paper it is shown that

X
¯

′
(
ε
)

\bar X\prime (\varepsilon )

is in fact constant when

ε
\varepsilon

lies outside a simply described finite set of rational numbers. More generally, let

k
k

be an algebraic number field and

S
S

finite set of absolute values of

k
k

containing the archimedean ones. For

υ
∈
S

\upsilon \in S

let

L
1
υ

,
…
,

L
m
υ

L_1^\upsilon , \ldots ,L_m^\upsilon

be linear forms with coefficients in

k
k

, and for

x
_

_

∈

K
n

∖
{

0
_

_

}

\underline {\underline x} \in {K^n}\backslash \{ \underline {\underline 0} \}

with height

H
k

(

x
_

_

)
>
1

{H_k}(\underline {\underline x} ) > 1

define

a

υ
i

(

x
_

_

)

{a_{\upsilon i}}(\underline {\underline x} )

by
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue upper L Subscript i Superscript upsilon Baseline left-parenthesis x underbar underbar right-parenthesis EndAbsoluteValue Subscript upsilon Baseline slash StartAbsoluteValue x underbar underbar EndAbsoluteValue Subscript upsilon Baseline equals upper H Subscript k Baseline left-parenthesis x underbar underbar right-parenthesis Superscript minus a Super Subscript upsilon i Superscript left-parenthesis x underbar underbar right-parenthesis slash d Super Subscript upsilon">

|

L
i
υ

(

x
_

_

)

|

υ

/

|

x
_

_

|

υ

=

H
k

(

x
_

_

)

−

a

υ
i

(

x
_

_

)

/

d
υ

|L_i^\upsilon (\underline {\underline x} )|_\upsilon /|\underline {\underline x} |_\upsilon = {H_k}{(\underline {\underline x} )^{ - {a_{\upsilon i}}(\underline {\underline x} )/{d_\upsilon }}}

where the

d
υ

{d_\upsilon }

are the local degrees. The approximation set

A
A

consists of tuples

a
_

_

=
{

a

υ
i

}

(
υ
∈
S
,
1
≦
i
≦
m
)

\underline {\underline a} = \{ {a_{\upsilon i}}\} \;(\upsilon \in S,1 \leqq i \leqq m)

such that for every neighborhood

O
O

of

a
_

_

\underline {\underline a}

the points

x
_

_

\underline {\underline x}

with

{

a

v
i

{

x
_

_

)
}
∈
O

\{ {a_{{v_i}}}\{ \underline {\underline x} )\} \in O

are dense in the subspace topology. Then

A
A

is a polyhedron whose vertices are rational points.