Let
A
=
{
a
s
+
n
s
Z
}
s
=
1
k
A= \{a_{s}+n_{s}\mathbb {Z}\}^{k}_{s=1}
(
n
1
⩽
⋯
⩽
n
k
)
n_{1} \leqslant \cdots \leqslant n_{k})
be a system of arithmetic sequences where
a
1
,
⋯
,
a
k
∈
Z
a_{1}, \cdots ,a_{k}\in \mathbb {Z}
and
n
1
,
⋯
,
n
k
∈
Z
+
n_{1},\cdots ,n_{k}\in \mathbb {Z}^{+}
. For
m
∈
Z
+
m\in \mathbb {Z}^{+}
system
A
A
will be called an (exact)
m
m
-cover of
Z
\mathbb {Z}
if every integer is covered by
A
A
at least (exactly)
m
m
times. In this paper we reveal further connections between the common differences in an (exact)
m
m
-cover of
Z
\mathbb {Z}
and Egyptian fractions. Here are some typical results for those
m
m
-covers
A
A
of
Z
\mathbb {Z}
: (a) For any
m
1
,
⋯
,
m
k
∈
Z
+
m_{1},\cdots ,m_{k}\in \mathbb {Z}^{+}
there are at least
m
m
positive integers in the form
Σ
s
∈
I
m
s
/
n
s
\Sigma _{s\in I} m_{s}/n_{s}
where
I
⊆
{
1
,
⋯
,
k
}
I \subseteq \{1,\cdots ,k\}
. (b) When
n
k
−
l
>
n
k
−
l
+
1
=
⋯
=
n
k
n_{k-l}>n_{k-l+1}= \cdots =n_{k}
(
0
>
l
>
k
)
0>l>k)
, either
l
⩾
n
k
/
n
k
−
l
l \geqslant n_{k}/n_{k-l}
or
Σ
s
=
1
k
−
l
1
/
n
s
⩾
m
\Sigma ^{k-l}_{s=1}1/n_{s} \geqslant m
, and for each positive integer
λ
>
n
k
/
n
k
−
l
\lambda >n_{k}/n_{k-l}
the binomial coefficient
(
l
λ
)
\binom l{ \lambda }
can be written as the sum of some denominators
>
1
>1
of the rationals
Σ
s
∈
I
1
/
n
s
−
λ
/
n
k
,
I
⊆
{
1
,
⋯
,
k
}
\Sigma _{s\in I}1/n_{s}- \lambda /n_{k}, I \subseteq \{1,\cdots ,k\}
if
A
A
forms an exact
m
m
-cover of
Z
\mathbb {Z}
. (c) If
{
a
s
+
n
s
Z
}
s
=
1
s
≠
t
k
\{a_{s}+n_{s}\mathbb {Z}\}^{k}_{\substack {s=1\ s\not =t}}
is not an
m
m
-cover of
Z
\mathbb {Z}
, then
Σ
s
∈
I
1
/
n
s
,
I
⊆
{
1
,
⋯
,
k
}
∖
{
t
}
\Sigma _{s\in I}1/n_{s}, I \subseteq \{1,\cdots ,k\}\setminus \{t\}
have at least
n
t
n_{t}
distinct fractional parts and for each
r
=
0
,
1
,
⋯
,
n
t
−
1
r=0,1,\cdots ,n_{t}-1
there exist
I
1
,
I
2
⊆
{
1
,
⋯
,
k
}
∖
{
t
}
I_{1},I_{2} \subseteq \{1,\cdots ,k\}\setminus \{t\}
such that
r
/
n
t
≡
Σ
s
∈
I
1
1
/
n
s
−
Σ
s
∈
I
2
1
/
n
s
r/n_{t} \equiv \Sigma _{s\in I_{1}}1/n_{s}-\Sigma _{s\in I_{2}}1/n_{s}
(mod 1). If
A
A
forms an exact
m
m
-cover of
Z
\mathbb {Z}
with
m
=
1
m=1
or
n
1
>
⋯
>
n
k
−
l
>
n
k
−
l
+
1
=
⋯
=
n
k
n_{1}> \cdots >n_{k-l}>n_{k-l+1}= \cdots =n_{k}
(
l
>
0
l>0
) then for every
t
=
1
,
⋯
,
k
t=1, \cdots ,k
and
r
=
0
,
1
,
⋯
,
n
t
−
1
r=0,1,\cdots ,n_{t}-1
there is an
I
⊆
{
1
,
⋯
,
k
}
I \subseteq \{1,\cdots ,k\}
such that
Σ
s
∈
I
1
/
n
s
≡
r
/
n
t
\Sigma _{s\in I}1/n_{s} \equiv r/n_{t}
(mod 1).