Abstract
AbstractLet $$f_1,\dots ,f_k\in \mathbb {R}[X]$$
f
1
,
⋯
,
f
k
∈
R
[
X
]
be polynomials of degree at most d with $$f_1(0)=\dots =f_k(0)=0$$
f
1
(
0
)
=
⋯
=
f
k
(
0
)
=
0
. We show that there is an $$n<x$$
n
<
x
such that $$\Vert f_i(n)\Vert _{\mathbb {R}/\mathbb {Z}}\ll x^{c/k}$$
‖
f
i
(
n
)
‖
R
/
Z
≪
x
c
/
k
for all $$1\le i\le k$$
1
≤
i
≤
k
for some constant $$c=c(d)$$
c
=
c
(
d
)
depending only on d. This is essentially optimal in the k-aspect, and improves on earlier results of Schmidt who showed the same result with $$c/k^2$$
c
/
k
2
in place of c/k.
Publisher
Springer Science and Business Media LLC
Subject
Geometry and Topology,Analysis
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