Let
K
\mathcal {K}
be the collection of all non-empty compact subsets of the unit cube
[
0
,
1
]
d
[0,1]^d
. Fixing any
F
σ
F_\sigma
set
F
⊂
[
0
,
1
]
d
F\subset [0,1]^d
of Lebesgue measure zero and
0
>
c
>
1
0>c>1
, we show that for typical
K
∈
K
K\in \mathcal {K}
with
L
d
(
K
)
≥
c
\mathcal {L}^d(K)\geq c
(in the sense of Baire category), the sumset
K
+
F
K+F
has empty interior. We also find the Hausdorff, lower and upper box and packing dimensions of
K
+
F
K+F
for typical
K
∈
K
K\in \mathcal {K}
.