123
Acta Math. Univ. Comenianae
Vol. LX, 1(1991), pp. 123–131
A THEOREM CONCERNING
SYSTEMS OF RESIDUE CLASSES
Z. W. SUN
We first introduce some notation. As usual (n1 , . . . , nk ) (resp. [n1 , . . . , nk ])
stands for the greatest common divisor (resp. least common multiple) of n1 , . . . , nk .
By system we mean a multi-set whose elements are unordered but may occur repeatedly. Following Š. Znám [8] we use a(n) to denote the residue class
{ x ∈ Z : x ≡ a (mod n) } .
For a system
A = {ai (ni )}ki=1
(1)
of residue classes, the ni are called its moduli.
Definition. An integer T is said to be a covering period of (1) if it is a period
Sk
of the characteristic function of the set i=1 ai (ni ) .
It is clear that [n1 , . . . , nk ] is a covering period of (1), and that any covering
period is a multiple of the smallest positive one.
For any set S of integers we use d(S) to denote the asymptotic density
1
|{ 0 ≤ x < N : x ∈ S }| .
N →∞ N
lim
(|A| is the cardinality of A.) The limit obviously exists if S is a union of finitely
many residue classes. In fact
d
k
[
!
ai (ni )
i=1
=
1
|{ 0 ≤ x < N : x ∈ ai (ni ) for some i }|
N
where N is any positive common multiple of n1 , . . . , nk .
Our main result is
Received November 10, 1989; revised June 21, 1990.
1980 Mathematics Subject Classification (1985 Revision). Primary 11825; Secondary 11A07,
11B05.
124
Z. W. SUN
Theorem. Let T be the smallest positive covering period of (1). Then we have
(n1 , . . . , nk )
≤ max |{ 1 ≤ i ≤ k : ni = n }|
(T, n1 , . . . , nk ) n∈Z+
(2)
X
[n ,... ,n ]
d| (n1 ,... ,nk )
1
1
.
d
k
To prove it we need two lemmas.
k
k
S
S
ai (ni ) ≥ d
0(ni ) .
Lemma 1. d
i=1
i=1
This is Lemma 2.3 of R.J.Simpson [6]. We can also prove it by using Theorem
1 of [2].
Lemma 2. Let n1 , . . . , nk ∈ Z+ , and let P be a finite set of primes such that
all the ni are contained in
P = { n ∈ Z+ : all prime divisors of n belong to P } .
Then
k
[
d
!
0(ni )
i=1

Y p−1

=
p

p∈P
X
k
1
.
n
n∈P ∩ ∪ 0(ni )
i=1
Proof. We note first that
Y
X 1
Y
1
1
p
1
1 + + 2 + ··· =
≤
=
.
n
n
p p
p−1
X
k
n∈P
n∈P ∩ ∪ 0(ni )
p∈P
p∈P
i=1
Let N = [n1 , . . . , nk ] and Nm =
Q m
p . For sufficiently large m we have
p∈P
N |Nm . From the inclusion-exclusion principle it follows that
d
!
k
[
0(ni )
i=1
=
1
N
1
=
N
(
)
k
[
0(ni ) 0≤x<N: x∈
i=1
k X
0 ≤ x < N : ni |x −
i=1
X 0 ≤ x < N : [ni , nj ]|x + · · ·
1≤i<j≤k
!
0 ≤ x < N : [n1 , . . . , nk ]|x k−1 + (−1)

k
1 X N
=
−
N i=1 ni
X
1≤i<j≤k

N
N

+ · · · + (−1)k−1
[ni , nj ]
[n1 , . . . , nk ]
A THEOREM CONCERNING SYSTEMS OF RESIDUE CLASSES
= Q p∈P
1
1+
1
p
+ (−1)k−1
=
Y
1−
p∈P
1
p
=
+ ···
d∈P
X1
1
[n1 , . . . , nk ]
d
X
1≤i<j≤k
X1
1
+ ···
[ni , nj ]
d
d∈P
!
d∈P
k
X
1 X 1
−
n Nm d
i=1 i
lim
m→∞
+ (−1)k−1

1
p2
+
k
X
1 X1
−
n
d
i=1 i
125
d|
1
[n1 , . . . , nk ]

ni
X
d| [n
Nm
1 ,... ,nk ]
1
d
X
1≤i<j≤k
1
[ni , nj ]
X
m
d| [nN,n
i
!
1
+ ···
d
j]
k
Y p−1
X
X 1
X
X
1
 lim
−
+ ···
m→∞
p
n
n
i=1 ni |n|Nm
p∈P
1≤i<j≤k [ni ,nj ]|n|Nm
!
X
1
k−1
+ (−1)
n
[n1 ,... ,nk ]|n|Nm
( d|n|m stands for “d|n and n|m ”)

Y p−1
 lim
=
m→∞
p

p∈P
X
ni |n|Nm
for some i


1 Y p − 1
=
n
p
p∈P
X
k
1
.
n
n∈ ∪ 0(ni )∩P
i=1
This concludes the proof.
Proof of Theorem. Since T is a covering period (1), we have
(
)
k
k
[
[
ai (ni ) = z + T y : z ∈
ai (ni ) and y ∈ Z
i=1
i=1
=
k
[
{ ai + ni x + T y : x, y ∈ Z } =
i=1
k
[
ai ((ni , T )) .
i=1
Let S denote the set {n1 , . . . , nk } and P be the set of all prime divisors of
[n1 , . . . , nk ]. Since
(n1 , . . . , nk )
[(n1 , . . . , nk ), T ]
=
(T, n1 , . . . , nk )
T
we have
(n1 , . . . , nk )
(T, n1 , . . . , nk )
and
n
i
(T, ni )
ni
[ni , T ]
=
,
(T, ni )
T
126
Z. W. SUN
and hence
ni
∈ 0((T, ni )) ∩ P .
(n1 , . . . , nk )/(T, n1 , . . . , nk )
Q δp
[n1 ,... ,nk ]
can be written in the form
p where δp ≥ 0. And it
Obviously, (n
1 ,... ,nk )
p∈P
is clear that
|ordp ni − ordp nj | ≤ ordp [n1 , . . . , nk ] − ordp (n1 , . . . , nk ) = δp .
(We use ordp n to denote the greatest integer α such that pα divides n.) So, if
n, n0 ∈ S and
Y
Y
pkp (1+δp ) = n0
plp (1+δp ) ,
n
p∈P
p∈P
then kp = lp for all p ∈ P and hence n = n0 .
Let M = max+ |{ 1 ≤ i ≤ k : ni = n }|. From Lemmas 1,2 and the above, we
n∈Z
have
!
k
k
k
X
X
[
1
=
d(ai (ni )) ≥ d
ai (ni ) = d
n
i=1 i
i=1
i=1

! 
k
[
Yp−1

≥d
0((ni , T )) = 
p
i=1
p∈P


k
[
!
ai ((ni , T ))
i=1
X
1
m
k
m∈ ∪ 0((ni ,T ))∩P
i=1
−1
n
p
(n1 , . . . , nk )/(T, n1 , . . . , nk )
p∈P
n∈S
Y
1
1
·
1 + 1+δp + 2(1+δ ) + · · ·
p
p
p
p∈P


Y
X1
p−1
1
(n1 , . . . , nk ) 

·
=
1
(T, n1 , . . . , nk )
p
1 − p1+δp n∈S n
p∈P


XM
(n1 , . . . , nk )  1 Y
pδp

=
δ
p
(T, n1 , . . . , nk ) M
1 + p + ···+ p
n
≥
Yp−1

X
p∈P
≥
(n1 , . . . , nk ) Y
1
·
M (T, n1 , . . . , nk )
1+
p∈P
Therefore
Y
(n1 , . . . , nk )
≤M
(T, n1 , . . . , nk )
p∈P
n∈S
1
p
1
+ ··· +
k
X
1
pδp i=1
1
1
1 + + · · · + δp = M
p
p
1
.
ni
X
[n ,... ,n ]
d| (n1 ,... ,nk )
1
k
1
,
d
127
A THEOREM CONCERNING SYSTEMS OF RESIDUE CLASSES
which is the desired result.
Remark 1. By checking the proof we see that (2) is implied by
k
X
1
≥d
n
i=1 i
(3)
k
[
!
0((ni , T ))
i=1
which holds if T is a covering period of (1).
We now say a few words about the theorem. If (n1 , . . . , nk )|T then (2) holds
trivially. Note that (2) can be written in the form
(20 )
1
≤ max |{ 1 ≤ i ≤ k : ni = n }|
(T, n1 , . . . , nk ) n∈Z+
X
(n1 ,... ,nk )|d|[n1 ,... ,nk ]
1
d
which is implied by
k
X
1
1
≥
.
n
(T,
n
,
. . . , nk )
i
1
i=1
(4)
If T |(n1 , . . . , nk ) then (4) holds, for
k
X
1
≥d
n
i=1 i
k
[
!
ai (ni )
≥ d(a1 (T )) =
i=1
1
.
(T, n1 , . . . , nk )
However (4) fails to hold in general, for example, the smallest positive covering
1
.
period of {0(2), 0(3)} is T = 6, but 12 + 13 6≥ (6,2,3)
Corollary. Let n0 be the smallest positive covering period of (1), and [n1 , . . . ,
nk ] have the prime factorization
[n1 , . . . , nk ] =
r
Y
i
pα
i ,
p1 < p2 < · · · < pr .
i=1
α
Suppose that pα
t 6 |n0 and pt |ns for some s = 1, . . . , k, and that ai (ni ) ∩ aj (nj ) = ∅
α
α
whenever pt |ni and pt 6 |nj (1 ≤ i, j ≤ k). Then we have
(5)
δ (α)
pt t
≤ εt (α) max |{ 1 ≤ i ≤ k : ni = ns }|
1≤s≤k
pα
t |ns
r
Y
pi
,
p −1
i=1 i
where
δt (α) = min{ δ ≥ 1 : ptα−δ kni
α
α
for some
(p kn stands for “p |n and p
α+1
0 ≤ i ≤ k}
6 |n” .)
128
Z. W. SUN
and
εt (α) =
Y
r 1
1 − αt −α+1
1 − αi +1 .
pt
pi
i=1
1
i6=t
Proof. Let I = { 1 ≤ i ≤ k : pα
t |ni } and J = {0, 1, . . . , k} − I. Obviously I 6= ∅,
0 ∈ J and pα
t 6 |nj for every j ∈ J. If i ∈ I and j ∈ J −{0} then ai (ni )∩aj (nj ) = ∅.
From this it follows that
x∈
[
ai (ni ) implies x ± [nj ]j∈J ∈
k
[
ai (ni ) −
i=1
i∈I
[
aj (nj ) =
[
ai (ni ) .
i∈I
j∈J−{0}
Hence the smallest positive covering period of {ai (ni )}i∈I must be a divisor of
[nj ]j∈J .
Applying the theorem we get
(ni )i∈I
≤ max |{ 1 ≤ i ≤ k : ni = ns }|
s∈I
((ni )i∈I , [nj ]j∈J )
(6)
X
[n ]
d| (ni )i∈I
1
.
d
i i∈I
(Notice that i ∈ I if 1 ≤ i ≤ k and ni = ns for some s ∈ I.) Since pα
t |(ni )i∈I we
have
ih
i
h
[nj ]j∈J , (ni )i∈I
[nj ]j∈J , pα
t
[nj ]j∈J
[nj ]j∈J
δ (α)
t
α
and thus the left side of (6) is a multiple of pα
t /(pt , [nj ]j∈J ) = pt
right side of (6), we note that
X
[n ]
d| (ni )i∈I
i i∈I
1
≤
d
X
d|
[ni ]i∈I
pα
t
1
≤
d
X
α −α
d|pt t
Qr
αi
i=1 pi
i6=t
. As for the
1
d
1
1
1
1
1
1+
+ 2 + · · · − αt −α+1 1 +
+ 2 + ···
pt p t
p t pt
pt
r
Y
1
1
1
1
1
·
1+
+ 2 + · · · − αi +1 1 +
+ 2 + ···
pi
pi
pi
pi
pi
i=1
=
i6=t
= εt (α)
r
Y
i=1
pi
.
pi − 1
Combining the above we obtain (5) from (6).
Remark 2. 1 ≤ δt (α) ≤ α, 0 < εt (α) < 1.
A THEOREM CONCERNING SYSTEMS OF RESIDUE CLASSES
129
Suppose that (1) is a disjoint system (i.e. a1 (n1 ), . . . , ak (nk ) are pairwise disr
joint). If pα
r does not divide (the smallest positive covering period) n0 , then by
the corollary we have
pδrr (αr ) ≤ max |{ 1 ≤ i ≤ k : ni = ns }|
(7)
1≤s≤k
r−1
Y
i=1
r
pα
r kns
pi
.
pi − 1
.) This is the first result announced in Sun [7].
(Note that εr (αr ) ≤ prp−1
r
Assume that each modulus of the disjoint system (1) occurs at most M times
(i.e.
|{ 1 ≤ i ≤ k : ni = ns }| ≤ M for every s = 1, . . . , k). By Merten’s theorem
(cf.[5]), we have
1 Y
p
ln x
∼ eγ
x p<x p − 1
x
where γ is the Euler constant ,
p prime
and thus
1 Y
p
1
<
x p<x p − 1
M
for sufficiently large x .
p prime
Let p∗ be the smallest prime such that
p∗ > M
Y
p<p∗
p prime
p
.
p−1
r
If pα
r 6 |n0 , in view of (7), we have
pr ≤ M
r−1
Y
i=1
Y
pi
p
≤M
,
pi − 1
p
−
1
p<p
r
p prime
and hence p∗ is an upper bound of prime divisors of n1 , . . . , nk . If pr ≥ p∗ we
r
must have pα
r kn0 .
Now let’s suppose the disjoint system (1) is also a covering, that is to say, (1)
is a disjoint covering system (i.e. ai (ni ), 1 ≤ i ≤ k, form a partition of Z). By the
corollary,
δ (α)
pt ≤ p t t
<M
r
Y
pi
p −1
i=1 i
for all t = 1, . . . , r and α = 1, . . . , αt .
(Notice that n0 = 1 and εt (α) < 1.) This establishes Burshtein’s conjecture ([4]).
r
Q
pi
(The original conjecture is that pr ≤ M
pi −1 .)
Let 1 ≤ t ≤ r,
i=1
δt = δt (αt ) = min{ δ ≥ 1 : ptαt −δ kni
for some 0 ≤ i ≤ k }
130
Z. W. SUN
and
Mt =


r

Q

 1 + pδt t




i=1
i6=t
pi −1 
pi
pδt t
if r > 1,
if r = 1.
([·] is the greatest integer function.) In [3] Berger, Felzenbaum and Fraenkel
showed that


r
Y
pi − 1 

M ≥ 1 + (pt − 1)
,
pi
i=1
i.e. pt
r
Y
pi − 1
i=1
i6=t
pi
<M .
In [6] R.J.Simpson proved that
M ≥ pr
r−1
Y
i=1
pi − 1
,
pi
and then he derived that there exists a number B(M ) such that, in any disjoint
covering system whose moduli are repeated at most M times, the least modulus
is less that B(M ). It is obvious that
M r ≥ pr
r−1
Y
i=1
pi − 1
.
pi
if r > 1, and εt (αt ) = ptp−1
if r = 1) we
Given 1 ≤ t ≤ r, (since εt (αt ) < ptp−1
t
t
t
have from the corollary M ≥ Mt , moreover there exists a modulus divided by pα
t
αt +1
and not by pt
which is repeated at least Mt times. If r ≥ 2 then
M r > pr
r−1
Y
i=1
and thus
r−2
Y pi − 1
pi − 1
p1 − 1
≥ pr−1
≥ · · · ≥ p2
pi
p
p1
i
i=1
M ≥ p2 (1 − p−1
1 ) +1 .
The last inequality was first proved by Berger, Felzenbaum and Fraenkel [1]. There
something was said about which modulus must occur at least [p2 (1 − p−1
1 )] + 1
times.
Acknowledgement. I am grateful to Prof. Štefan Znám for his encouraging
me to publish this paper.
A THEOREM CONCERNING SYSTEMS OF RESIDUE CLASSES
131
References
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systems of residue sets, Ars. Combin. 20 (1985), 69–82.
2.
, The Herzog-Schonheim conjecture for finite nilpotent groups, Canad. Math. Bull. 29
(1986), 329–333.
3.
, Lattice Paralleloptopes and disjoint covering systems, Discrete Math. 65 (1987),
23–46.
4. Burshtein N., On natural exactly covering systems of congruences having moduli occurring at
most M times, Discrete Math. 14 (1976), 205–214.
5. Hardy G. H. and Wright E. M., An Introduction to the Theory of Numbers, 5th ed., Oxford
Univ. Press, 1981.
6. Simpson R. J., Exact coverings of the integers by arithmetic progressions, Discrete Math. 59
(1986), 181–190.
7. Sun Z. W., Several results on systems of residue classes, Adv. in Math. (Beijing), No. 2 18
(1989), 251–252.
8. Znám Š., On covering sets of residue classes, in: P. Turán, ed., Topics in Number Theory, Colloq. Math., Societatis, János Bolyai Vol. 13, Debrecen, 1974, (North-Holland, 1976),
443–449.
Z. W. Sun, Department of Mathematics, Nanjing University, Nanjing 210008, People’s Republic
of China