Let
G
G
be a Vilenkin group of bounded order and
H
n
{H_n}
a sequence of clopen subgroups of
G
G
forming a base at the identity. If
E
E
is a subset of
G
G
, let
N
n
(
E
)
{N_n}(E)
denote the number of cosets of
H
n
{H_n}
which intersect
E
E
. If
\[
lim
_
N
n
(
E
)
log
[
G
:
H
n
]
>
∞
,
\underline {\lim } \frac {{{N_n}(E)}} {{\log [G:{H_n}]}} > \infty ,
\]
then
E
E
is a U-set in the group
G
G
. It is also shown that for
G
G
satisfying a growth condition and
φ
(
n
)
→
∞
\varphi (n) \to \infty
, there is an M-set,
E
E
, with
\[
N
n
(
E
)
=
O
(
φ
(
n
)
log
[
G
:
H
n
]
)
.
{N_n}(E) = O(\varphi (n)\,\log [G:{H_n}]).
\]