Theorem. If
X
1
,
…
,
X
n
X_{1},\dots ,X_{n}
are perfect compact subsets of the locally compact metrizable abelian group, then there are pairwise disjoint perfect subsets
Y
1
⊆
X
1
,
…
,
Y
n
⊆
X
n
Y_{1}\subseteq X_{1},\dots ,Y_{n}\subseteq X_{n}
such that (i)
Y
j
Y_{j}
is either a Kronecker set or (ii) for some
p
j
≥
2
p_{j}\ge 2
,
Y
j
Y_{j}
is a translate of a
K
p
j
K_{p_{j}}
-set all of whose elements have order
p
j
p_{j}
, and (iii)
A
(
Y
1
+
⋯
+
Y
n
)
A(Y_{1}+\dots +Y_{n})
is isomorphic to the projective tensor product
C
(
Y
1
)
⊗
^
⋯
⊗
^
C
(
Y
n
)
C(Y_{1}) \hat \otimes \cdots \hat \otimes C(Y_{n})
. This extends what was previously known for groups such as
T
\mathbb {T}
or for the case
n
=
2
n=2
to the general locally compact abelian group. Old results concerning the local existence of Kronecker and
K
p
K_{p}
-sets are improved.