LXXXI.2 (1997)

**Exact m-covers and the linear form**^{P}^{k}_{s=1}*x**s**/n**s*
by

Zhi-Wei Sun (Nanjing)

**1. Introduction. For a, n ∈ Z with n > 0, we let**

*a + nZ = {. . . , a − 2n, a − n, a, a + n, a + 2n, . . .}*

*and call it an arithmetic sequence. Given a finite system*

(1) *A = {a*_{s}*+ n*_{s}*Z}*^{k}_{s=1}

*of arithmetic sequences, we assign to each x ∈ Z the corresponding covering*
*multiplicity σ(x) = |{1 ≤ s ≤ k : x ∈ a*_{s}*+ n*_{s}*Z}| (|S| means the cardinality*
*of a set S), and call m(A) = inf*_{x∈Z}*σ(x) the covering multiplicity of A.*

Apparently (2)

X*k*
*s=1*

1
*n**s* = 1

*N*

*N −1*X

*x=0*

*σ(x) ≥ m(A)*

*where N is the least common multiple of those common differences (or*
*moduli) n*_{1}*, . . . , n*_{k}*. For a positive integer m, (1) is said to be an m-cover*
*of Z if its covering multiplicity is not less than m, and an exact m-cover*
*of Z if σ(x) = m for all x ∈ Z. Note that k ≥ m if (1) forms an m-cover*
*of Z. Clearly the covering function σ : Z → Z is constant if and only if (1)*
*forms an exact m-cover of Z for some m = 1, 2, . . . An exact 1-cover of Z is*
a partition of Z into residue classes.

*P. Erd˝os ([E]) proposed the concept of cover (i.e., 1-cover) of Z in the*
1930’s, ˇ*S. Porubsk´y ([P]) introduced the notion of exact m-cover of Z in*
*the 1970’s, and the author ([Su3]) studied m-covers of Z for the first time.*

*The most challenging problem in this field is to describe those n*_{1}*, . . . , n*_{k}*in an m-cover (or exact m-cover) (1) of Z (cf. [Gu]). In [Su2, Su3, Su4]*

*the author revealed some connections between (exact) m-covers of Z and*

*1991 Mathematics Subject Classification: Primary 11B25; Secondary 11A07, 11B75,*
11D68.

This research is supported by the National Natural Science Foundation of P.R. China.

[175]

*Egyptian fractions. Here we concentrate on exact m-covers of Z. In [Su3,*
*Su4] results for exact m-covers of Z were obtained by studying general*
*m-covers of Z and noting that an exact m-cover (1) of Z is an m-cover*
of Z with P_{k}

*s=1**1/n*_{s}*= m. In Section 4 of the present paper we shall di-*
*rectly characterize exact m-covers of Z in various ways. (Note that in the*
famous book [Gu] R. K. Guy wrote that characterizing exact 1-covers of Z is
a main outstanding unsolved problem in the area.) This enables us to make
further progress. With the help of the linear form P_{k}

*s=1**x**s**/n**s* (studied in
*the next section), we will provide some new properties of exact m-covers of Z*
(see Section 3). The fifth section is devoted to proofs of the main theorems
stated in Section 3.

*For a complex number x and nonnegative integer n, as usual,*

*x*
*n*

:= 1

*n!*

*n−1*Y

*j=0*

*(x − j)*

*x*
0

is 1

*. For real x we use [x] and {x} to represent the integral part and*
*the fractional part of x respectively. For two integers a, b not both zero,*
*(a, b) denotes the greatest common divisor of a and b.*

*Now we state our central results for an exact m-cover (1) of Z:*

*(I) For a = 0, 1, 2, . . . and t = 1, . . . , k there are at least* _{[a/n}^{m−1}

*t*]

*subsets I*
*of {1, . . . , k} for which t 6∈ I and*P

*s∈I**1/n*_{s}*= a/n** _{t}*, where the lower bounds
are best possible.

*(II) If ∅ 6= I ⊆ {1, . . . , k} and (n*_{s}*, n*_{t}*) | a*_{s}*− a*_{t}*for all s, t ∈ I, then*

X

*s∈J*

1
*n*_{s}

*: J ⊆ {1, . . . , k} \ I*

*⊇*

*r*

*[n** _{s}*]

_{s∈I}*: r = 0, 1, . . . , [n*

*s*]

*s∈I*

*− 1*

*where [n**s*]*s∈I* *is the least common multiple of those n**s* *with s ∈ I.*

*(III) For any rational c, the number of solutions of the equation*
P_{k}

*s=1**x*_{s}*/n*_{s}*= c with x*_{s}*∈ {0, 1, . . . , n*_{s}*− 1} for s = 1, . . . , k, is the sum*
*of finitely many (not necessarily distinct) prime factors of n*1*, . . . , n**k* if
*c 6= 0, 1, 2, . . . , and at least* ^{k−m}_{n}

*if c equals a nonnegative integer n.*

**2. On the linear form** P_{k}

*s=1**x*_{s}*/n*_{s}**. In this section we shall say some-**
thing general about the linear formP_{k}

*s=1**x*_{s}*/n*_{s}*where n*_{1}*, . . . , n** _{k}*are positive
integers.

*Let us first introduce more notations. For x, y in the rational field Q,*
*if x − y ∈ Z then we write x ≡ y (mod 1). For n = 1, 2, . . . we set R(n) =*
*{0, . . . , n − 1}. When we deal with a finite collection {n*_{s}*}** _{s∈I}* of positive

*integers, the least common multiple [n*

*s*]

*s∈I*and the product Q

*s∈I**n**s* will
*be regarded as 1 if I is empty.*

*Definition. Two (finite) sequences {n**s**}*^{k}_{s=1}*and {m**t**}*^{l}* _{t=1}* of positive

*integers are said to be equivalent if k = l and (n*

_{s}*, n*

_{t}*) = (m*

_{s}*, m*

*) for all*

_{t}*s, t = 1, . . . , k with s 6= t. We call {n*

_{s}*}*

^{k}

_{s=1}*a normal sequence if n*

*divides*

_{t}*[n*

*s*]

^{k}

_{s=1, s6=t}*for every t = 1, . . . , k.*

*Proposition 2.1. Let n*1*, . . . , n**k* *be arbitrary positive integers. Then*
*{(n*_{t}*, [n** _{s}*]

^{k}

_{s=1, s6=t}*)}*

^{k}

_{t=1}*is the only normal sequence equivalent to {n*

_{s}*}*

^{k}

_{s=1}*.*

*P r o o f. For each t = 1, . . . , k we let*

*n*^{0}_{t}*= (n*_{t}*, [n** _{s}*]

^{k}

_{s=1, s6=t}*) = [(n*

_{s}*, n*

*)]*

_{t}

^{k}

_{s=1, s6=t}*.*

*Clearly n*^{0}_{t}*divides [n*^{0}* _{s}*]

^{k}

_{s=1, s6=t}*because (n*

_{s}*, n*

_{t}*) | n*

^{0}

_{s}*for all s = 1, . . . , k with*

*s 6= t. For i, j = 1, . . . , k with i 6= j, (n*

^{0}

_{i}*, n*

^{0}

_{j}*) = (n*

_{i}*, n*

_{j}*) since n*

_{i}*| [n*

*]*

_{s}

^{k}

_{s=1, s6=j}*and n*

*j*

*| [n*

*s*]

^{k}

_{s=1, s6=i}*. Hence {n*

^{0}

_{s}*}*

^{k}

_{s=1}*is normal and equivalent to {n*

*s*

*}*

^{k}*. If*

_{s=1}*so is {m*

_{s}*}*

^{k}

_{s=1}*where m*

_{1}

*, . . . , m*

*are positive integers, then*

_{k}*m**t**= (m**t**, [m**s*]^{k}_{s=1, s6=t}*) = [(m**s**, m**t*)]^{k}_{s=1, s6=t}*= [(n**s**, n**t*)]^{k}_{s=1, s6=t}*= n*^{0}_{t}*for every t = 1, . . . , k. We are done.*

*Proposition 2.2. Let n*_{1}*, . . . , n*_{k}*be positive integers. For θ ∈ Q the*
*equation*

(3)

X*k*
*s=1*

*x**s*

*n*_{s}*≡ θ (mod 1)* *with x**s**∈ R(n**s**) for s = 1, . . . , k*

*is solvable if and only if [n*_{1}*, . . . , n*_{k}*]θ ∈ Z, and in the solvable case the*
*number of solutions is n*_{1}*. . . n*_{k}*/[n*_{1}*, . . . , n*_{k}*], which does not change if we*
*replace {n**s**}*^{k}_{s=1}*by an equivalent sequence.*

*P r o o f. We argue by induction. The case k = 1 is trivial. Let k > 1 and*
*assume Proposition 2.2 for smaller values of k. Observe that*

1

*[n*1*, . . . , n**k*]Z = *([n*1*, . . . , n**k−1**], n**k*)
*[n*1*, . . . , n**k−1**]n**k* Z = 1

*n**k*Z + 1

*[n*1*, . . . , n**k−1*]*Z.*

*So [n*1*, . . . , n**k**]θ ∈ Z if and only if [n*1*, . . . , n**k−1**](θ − x/n**k**) ∈ Z for some*
*x ∈ Z. For any a ∈ Z with 0 ≤ a < n** _{k}*, the congruence

*k−1*X

*s=1*

*x*_{s}

*n*_{s}*≡ θ −* *a*

*n** _{k}* (mod 1)
is solvable if and only if

*[n*_{1}*, . . . , n** _{k−1}*]

*θ −* *a*

*n*_{k}

*∈ Z,*
i.e.

*[n*_{1}*, . . . , n*_{k−1}*]a ≡ [n*_{1}*, . . . , n*_{k−1}*]n*_{k}*θ (mod n*_{k}*).*

*Hence (3) is solvable if and only if [n*1*, . . . , n**k**]θ ∈ Z. In the solvable case*
*there are exactly ([n*_{1}*, . . . , n*_{k−1}*], n*_{k}*) = [(n*_{1}*, n*_{k}*), . . . , (n*_{k−1}*, n** _{k}*)] numbers

*a ∈ R(n*

*) satisfying the last congruence, thus by the induction hypothesis (3) has exactly*

_{k}*n*1*. . . n**k−1*

*[n*_{1}*, . . . , n** _{k−1}*]

*([n*

_{1}

*, . . . , n*

_{k−1}*], n*

*) =*

_{k}*n*1

*. . . n*

*k*

*[n*_{1}*, . . . , n** _{k}*]

*solutions. As n*1*. . . n**k−1**/[n*1*, . . . , n**k−1**] depends only on those (n**i**, n**j*) with
*1 ≤ i < j < k, the number n*_{1}*. . . n*_{k}*/[n*_{1}*, . . . , n** _{k}*] depends only on the

*(n*

*s*

*, n*

*t*

*), 1 ≤ s < t ≤ k. This ends the proof.*

*Corollary 2.1. Let a be an integer and n*_{1}*, . . . , n*_{k}*positive integers.*

*Then a/[n*_{1}*, . . . , n*_{k}*] can be written uniquely in the form q +*P_{k}

*s=1**x*_{s}*/n*_{s}*with q ∈ Z and x**s* *∈ R(n**s**) for s = 1, . . . , k if and only if (n**s**, n**t**) = 1 for all*
*s, t = 1, . . . , k with s 6= t.*

*P r o o f. By Proposition 2.2, equation (3) with θ = a/[n*1*, . . . , n**k*] has a
*unique solution if and only if n*_{1}*. . . n*_{k}*= [n*_{1}*, . . . , n** _{k}*]. So the desired result
follows.

*Corollary 2.2. Let n*1*, . . . , n**k* *be positive integers. Then the number of*
*solutions of the equation*

(4)
X*k*
*s=1*

*x**s*

*n**s* *≡ 0 (mod 1)* *with x**s**∈ Z and 0 < x**s* *< n**s* *for s = 1, . . . , k*
*equals*

*(−1)** ^{k}*+
X

*k*

*t=1*

*(−1)** ^{k−t}* X

*1≤i*1*<...<i**t**≤k*

*n**i*_{1}*. . . n**i*_{t}

*[n*_{i}_{1}*, . . . , n*_{i}* _{t}*]

*which depends only on those (n*

_{s}*, n*

_{t}*) with 1 ≤ s < t ≤ k.*

*P r o o f. For I ⊆ {1, . . . , k} let #I denote the number of solutions of the*
diophantine equation P

*s∈I**x*_{s}*/n*_{s}*≡ 0 (mod 1) with x*_{s}*∈ {1, . . . , n*_{s}*− 1}*

*for s ∈ I, and consider #∅ to be 1. By Proposition 2.2,* P

*J⊆I**#J =*
Q

*s∈I**n**s**/[n**s*]*s∈I* *for all I ⊆ {1, . . . , k}, therefore #{1, . . . , k} coincides with*
X

*J⊆{1,...,k}*

*k−|J|*X

*s=0*

*(−1)*^{k−|J|−s}

*k − |J|*

*s*

*#J*

= X

*J⊆{1,...,k}*

X

*J⊆I⊆{1,...,k}*

*(−1)*^{k−|I|}*#J*

= X

*I⊆{1,...,k}*

*(−1)** ^{k−|I|}*X

*J⊆I*

*#J =* X

*I⊆{1,...,k}*

*(−1)*^{k−|I|}

Q

*s∈I**n**s*

*[n** _{s}*]

_{s∈I}*= (−1)** ^{k}*+
X

*k*

*t=1*

*(−1)** ^{k−t}* X

*1≤i*1*<...<i**t**≤k*

*n*_{i}_{1}*. . . n*_{i}_{t}*[n*_{i}_{1}*, . . . , n*_{i}* _{t}*]

*.*

*In view of Proposition 2.2, the number #{1, . . . , k} remains the same if an*
*equivalent sequence is substituted for {n*_{s}*}*^{k}* _{s=1}*. The proof is now complete.

R e m a r k 1. Equation (4) is closely related to diagonal hypersurfaces
over a finite field. The formula for the number of solutions of (4) was ob-
tained by R. Lidl and H. Niederreiter [LN], R. Stanly (cf. C. Small [Sm]),
Q. Sun, D.-Q. Wan and D.-G. Ma [SWM] with much more complicated
*methods. The fact that the number does not vary if we replace {n**s**}*^{k}* _{s=1}*
by the corresponding normal sequence, was recently noted by A. Granville,
S.-G. Li and Q. Sun [GLS]. For necessary and sufficient conditions for the
solvability of (4), the reader is referred to [SW] where the authors deter-
mined when (4) has a unique solution.

*Corollary 2.3. Let (1) be a system of arithmetic sequences with*
*(n**s**, n**t**) | a**s**− a**t* *for all s, t = 1, . . . , k. Then for any θ ∈ Q with 0 ≤ θ < 1*
*we have*

(5)

X

*x*_{s}*∈R(n** _{s}*)

*for*

*s=1,...,k*

*{*P

_{k}*s=1**x*_{s}*/n*_{s}*}=θ*

*e*^{2πi}^{P}^{k}^{s=1}^{a}^{s}^{x}^{s}^{/n}^{s}

=

*( n*_{1}*. . . n*_{k}

*[n*_{1}*, . . . , n** _{k}*]

*if [n*

_{1}

*, . . . , n*

_{k}*]θ ∈ Z,*

0 *otherwise.*

P r o o f. By the Chinese Remainder Theorem in general form, the inter-
section T_{k}

*s=1**a*_{s}*+ n*_{s}*Z is nonempty if and only if a*_{s}*+ n*_{s}*Z ∩ a*_{t}*+ n*_{t}*Z 6= ∅*
*for all s, t = 1, . . . , k. (For a proof see, e.g., [Su1].) Since (n**s**, n**t**) | a**s**− a**t* for
*s, t = 1, . . . , k,* T_{k}

*s=1**a*_{s}*+ n*_{s}*Z must contain an integer x. With the help of*
Proposition 2.2,

X

*x**s**∈R(n**s*)for*s=1,...,k*
*{*P_{k}

*s=1**x**s**/n**s**}=θ*

*e*^{2πi}^{P}^{k}^{s=1}^{a}^{s}^{x}^{s}^{/n}* ^{s}* = X

*x**s**∈R(n**s*)for*s=1,...,k*
*{*P_{k}

*s=1**x**s**/n**s**}=θ*

*e*^{2πixθ}

*vanishes if [n*_{1}*, . . . , n*_{k}*]θ 6∈ Z, and otherwise equals* _{[n}^{n}^{1}^{...n}^{k}

1*,...,n** _{k}*]

*e*

*. So (5) holds.*

^{2πixθ}To conclude this section we make a few comments. For system (1),
*M (A) = sup*_{x∈Z}*σ(x) does not change if an equivalent sequence takes the*
*place of {n**s**}*^{k}_{s=1}*, because for ∅ 6= I ⊆ {1, . . . , k} the set* T

*s∈I**a**s**+ n**s*Z is
*nonempty if and only if (n*_{s}*, n*_{t}*) | a*_{s}*− a*_{t}*for all s, t ∈ I. Observe that (1)*

*forms an exact m-cover of Z if and only if* P_{k}

*s=1**1/n**s* *= m ≥ M (A). So*
*whether n*_{1}*, . . . , n*_{k}*are the moduli of an exact m-cover of Z only depends on*
P_{k}

*s=1**1/n*_{s}*and the k(k − 1)/2 numbers (n*_{s}*, n*_{t}*), 1 ≤ s < t ≤ k. For a given*
*exact m-cover (1) of Z, replacing {n**s**}*^{k}* _{s=1}* by the unique normal sequence

*{n*

^{0}

_{s}*}*

^{k}*equivalent to it we find that*

_{s=1}X*k*
*s=1*

1

*n*^{0}_{s}*≤ M (A) ≤ m =*
X*k*
*s=1*

1
*n*_{s}*.*

*As n*^{0}_{s}*≤ n**s* *for s = 1, . . . , k, the sequence {n**s**}*^{k}* _{s=1}* must be identical with

*{n*

^{0}

_{s}*}*

^{k}*and hence normal. In the light of the above, the reader should not*

_{s=1}*be surprised by connections between the exact m-cover (1) of Z and the*linear formP

_{k}*s=1**x**s**/n**s**.*

**3. Main theorems and their consequences. In this section we let (1)**
*be an exact m-cover of Z; we also let I ⊆ {1, . . . , k} and ¯I = {1, . . . , k} \ I.*

*For any rational c, we let I*^{∗}*(c) be the number of solutions hx**s**i**s∈I* to the
diophantine equation

(6) X

*s∈I*

*x*_{s}*n**s*

*= c* *with x*_{s}*∈ R(n*_{s}*) for all s ∈ I,*
*and I**∗**(c) = |{J ⊆ I :*P

*s∈J**1/n**s* *= c}| be the number of solutions hδ**s**i**s∈I*

to the equation

(7) X

*s∈I*

*δ**s*

*n**s* *= c* *with δ**s**∈ R(2) = {0, 1} for all s ∈ I.*

*(When I = ∅ and c = 0 we view each of (6) and (7) as having only the zero*
solution.) We also set

*I**∗*^{(0)}*(c) =*

*J ⊆ I : 2 | |J| and* X

*s∈J*

1
*n*_{s}*= c*

(8)

and

*I*_{∗}^{(1)}*(c) =*

*J ⊆ I : 2 - |J| and* X

*s∈J*

1
*n**s*

*= c*

(9) *.*

Let us present our main theorems whose proofs will be given later, and derive a number of interesting corollaries from them.

*Theorem 3.1. Let c be a rational number.*

*(i) When |I| ≤ m, if I*^{∗}*(c − n) = 1 for a nonnegative integer n then*
(10) *I*¯_{∗}*(c) +*

*m−|I|*X

*l=0**l6=n*

*m − |I|*

*l*

*I*^{∗}*(c − l) ≥*

*m − |I|*

*n*

;

*in particular , if c can be uniquely written in the form n +*P

*s∈I**x**s**/n**s* *where*
*n and x*_{s}*lie in {0, 1, . . . , m − |I|} and {0, 1, . . . , n*_{s}*− 1} respectively, then*

*I*¯_{∗}*(c) ≥*

*m − |I|*

*n*

*.*

*(ii) When |I| ≥ m, if ¯I**∗**(c − n) = 1 for a nonnegative integer n then*
(11) *I*^{∗}*(c) +*

*|I|−m*X

*l6=n**l=0*

*|I| − m*
*l*

*I*¯_{∗}*(c − l) ≥*

*|I| − m*
*n*

;

*in particular , if c can be uniquely expressed in the form n+*P

*s∈J**1/n*_{s}*where*
*J ⊆ ¯I and n ∈ {0, 1, . . . , |I| − m}, then*

*I*^{∗}*(c) ≥*

*|I| − m*
*n*

*.*

*Below there are corollaries involving the cases |I| ≤ m, |I| = m and*

*|I| ≥ m.*

*Corollary 3.1. Assume that those n*_{s}*with s ∈ I are pairwise relatively*
*prime. Then |I| ≤ m and*

(12)

*J ⊆ ¯I :*X

*s∈J*

1

*n**s* *= n +*X

*s∈I*

*x*_{s}*n**s*

* ≥*

*m − |I|*

*n*

*for all n = 0, 1, 2, . . . and x*_{s}*∈ R(n*_{s}*) with s ∈ I; in particular ,*
(13) X

*s∈J*

1

*n*_{s}*: J ⊆ ¯I*

*⊇*

*a*

*[n** _{s}*]

_{s∈I}*: a ∈ Z & |I| ≤*

*a*

*[n** _{s}*]

_{s∈I}*≤ m − |I|*

*and*
(14)

*J ⊆ ¯I :*X

*s∈J*

1

*n*_{s}*≡* *a*

Q

*s∈I**n** _{s}* (mod 1)

* ≥ 2*^{m−|I|}*for every a ∈ Z.*

P r o o f. By the Chinese Remainder Theorem,T

*s∈I**a**s**+n**s**Z 6= ∅ if I 6= ∅.*

*Since any integer lies in exactly m members of (1), |I| does not exceed m.*

*Let N = [n** _{s}*]

*=Q*

_{s∈I}*s∈I**n*_{s}*. By Corollary 2.1, for each a ∈ Z the number*
*a/N can be expressed uniquely in the form q +*P

*s∈I**x**s**/n**s* *with q ∈ Z and*
*x*_{s}*∈ R(n*_{s}*) for s ∈ I. Whenever x*_{s}*∈ R(n*_{s}*) for all s ∈ I, by Theorem 3.1,*
*(12) holds for every nonnegative integer n. If |I|N ≤ a ≤ (m−|I|)N then the*
*corresponding integer q = a/N −*P

*s∈I**x**s**/n**s* *lies in the interval [0, m − |I|]*

and hence

*J ⊆ ¯I :*X

*s∈J*

1
*n** _{s}* =

*a*

*N* *= q +*X

*s∈I*

*x**s*

*n*_{s}

* ≥*

*m − |I|*

*q*

*> 0.*

This yields (13). For (14) we observe that

*J ⊆ ¯I :*X

*s∈J*

1
*n**s* *≡* *a*

*N* (mod 1)

*≥*

*m−|I|*X

*n=0*

*J ⊆ ¯I :*X

*s∈J*

1

*n*_{s}*= n +*X

*s∈I*

*x*_{s}*n*_{s}

*≥*

*m−|I|*X

*n=0*

*m − |I|*

*n*

= 2^{m−|I|}*.*
This concludes the proof.

*Applying Corollary 3.1 with I = ∅ we immediately get the theorem of*
Sun [Su2].

*Putting I = {t} (1 ≤ t ≤ k) in Corollary 3.1 we then obtain result (I)*
*stated in the first section. In the case m = 1, result (I) was first observed*
*by the author in [Su4]. When m > 1, we noted in [Su4] that, providing*
*n*1 *< . . . < n**k−l* *< n**k−l+1* *= . . . = n**k**, for every r = 0, 1, . . . , n**k**− 1 there*
*exists a J ⊆ {1, . . . , k − 1} with* P

*s∈J**1/n*_{s}*≡ r/n*_{k}*(mod 1). In [Su4] we*
*even conjectured that, if (1) forms an m-cover of Z with σ(x) = m for all*
*x ≡ a**t* *(mod n**t**) where 1 ≤ t ≤ k, then*

(15) X

*s∈I*

1
*n**s*

*: I ⊆ {1, . . . , k} \ {t}*

*∩* 1
*n**t*Z

=

*r*

*n*_{t}*: r = 0, . . . , n**t**− 1*

*.*
*Result (I) confirms the conjecture for exact m-covers of Z. The lower bounds*
are best possible as is shown by the following example.

*Example. Let k > m > 0 be integers. Let a**s* *= 0 and n**s* = 1 for
*s = 1, . . . , m − 1, a** _{s}* = 2

^{s−m}*and n*

*= 2*

_{s}

^{s−m+1}*for s = m, . . . , k − 1, also*

*a*

_{k}*= 0 and n*

*= 2*

_{k}

^{k−m}*. It is clear that A = {a*

_{s}*+ n*

_{s}*Z}*

^{k}*forms an exact*

_{s=1}*m-cover of Z. As each nonnegative integer can be expressed uniquely in*

*the binary form, the reader can easily check that for a = 0, 1, 2, . . . and*

*t = 1, . . . , k we always have*

*J ⊆ {1, . . . , k} \ {t} :*X

*s∈J*

1
*n** _{s}* =

*a*

*n*_{t}

=

*m − 1*
*[a/n** _{t}*]

*.*

*Corollary 3.2. Suppose that |I| = m. Then no number occurs exactly*
*once among the 2*^{k−m}*n*_{1}*. . . n*_{m}*rationals*

(16) X

*s∈I*

*x**s*

*n*_{s}*,* *x*_{s}*∈ R(n*_{s}*) for s ∈ I;* X

*s∈J*

1

*n*_{s}*,* *J ⊆ ¯I.*

*P r o o f. If I** ^{∗}*(P

*s∈I**x**s**/n**s**) = 1 where x**s* *∈ R(n**s**) for s ∈ I then*
*I*¯* _{∗}*(P

*s∈I**x*_{s}*/n*_{s}*) ≥* ^{m−|I|}_{0}

*= 1 by Theorem 3.1(i). If J ⊆ ¯I and*
*I*¯* _{∗}*(P

*s∈J**1/n*_{s}*) = 1, then I** ^{∗}*(P

*s∈J**1/n*_{s}*) ≥* ^{|I|−m}_{0}

= 1 by Theorem 3.1(ii).

We are done.

*Corollary 3.3. Assume that |I| ≥ m. For any J ⊆ ¯I, if*
(17)

X

*s∈J*^{0}

1

*n*_{s}*−*X

*s∈J*

1
*n*_{s}

* ∈ {0, 1, . . . , |I| − m}* *for no J*^{0}*⊆ ¯I with J*^{0}*6= J,*

*then*

(18) *I*^{∗}

*n +*X

*s∈J*

1
*n**s*

*≥*

*|I| − m*
*n*

*for n = 0, 1, 2, . . .*
*and hence*

(19) Y

*s∈I*

*n**s* *≥ 2*^{|I|−m}*[n**s*]*s∈I**.*

*P r o o f. Let J be a subset of ¯I which satisfies (17). Note that* ^{|I|−m}_{n}

= 0
*for every integer n > |I| − m. For n ∈ Z with 0 ≤ n ≤ |I| − m, if J*^{0}*⊆ ¯I and*
*n*^{0}*∈ {0, 1, . . . , |I| − m} then by (17),*

*n +*X

*s∈J*

1

*n*_{s}*= n** ^{0}*+ X

*s∈J*^{0}

1

*n*_{s}*⇒ J = J*^{0}*and n = n*^{0}*.*

So (18) holds in view of the latter part of Theorem 3.1, and thus by Propo- sition 2.2,

Q

*s∈I**n*_{s}*[n** _{s}*]

_{s∈I}*≥*

*hx*_{s}*i*_{s∈I}*: x*_{s}*∈ R(n*_{s}*) for s ∈ I &* X

*s∈I*

*x*_{s}*n*_{s}*≡*X

*s∈J*

1

*n** _{s}* (mod 1)

*≥*

*|I|−m*X

*n=0*

*I*^{∗}

*n +*X

*s∈J*

1
*n*_{s}

*≥*

*|I|−m*X

*n=0*

*|I| − m*
*n*

= 2^{|I|−m}*.*

*Putting I = {1, . . . , k} and J = ∅ in Corollary 3.3 we obtain the second*
*half of result (III). When 1 ≤ t ≤ k and n*_{t}*> 1, Corollary 3.3 in the case*
*I = {1, . . . , k} \ {t} and J = {t} also yields an interesting result.*

*Let p(1) = 1 and p(n) denote the smallest (positive) prime factor of n*
*for n = 2, 3, . . . For a positive integer n we also put*

(20) *D(n) =*n X

*p | n*

*pm**p**: all the m**p* are nonnegative integers
o

*.*

*Theorem 3.2. Let c be a rational number.*

*(i) If |I| ≤ m, then either*
(21) *I*¯_{∗}*(c) +*

*m−|I|*X

*n=0*

*I*^{∗}*(c − n) ≥ p([n*_{1}*, . . . , n** _{k}*])

*or*

(22) *I*¯_{∗}^{(0)}*(c) − ¯I*_{∗}^{(1)}*(c) =*

*m−|I|*X

*n=0*

*(−1)*^{n}

*m − |I|*

*n*

*I*^{∗}*(c − n);*

*moreover*

(23) *I*¯*∗**(c) +*

*m−|I|*X

*n=0*

*m − |I|*

*n*

*I*^{∗}*(c − n) ∈ D([n*1*, . . . , n**k*])
*if |S|, |T | ≤ 1 and S ∩ T = ∅ where*

*S = {n mod 2 : n ∈ Z, 0 ≤ n ≤ m − |I| and I*^{∗}*(c − n) 6= 0}*

*and*

*T =*

*|J| mod 2 : J ⊆ ¯I and* X

*s∈J*

1
*n*_{s}*= c*

*.*
*(ii) If |I| ≥ m, then either*

(24) *I*^{∗}*(c) +*

*|I|−m*X

*n=0*

*I*¯_{∗}*(c − n) ≥ p([n*_{1}*, . . . , n** _{k}*])

*or*

(25) *I*^{∗}*(c) =*

*|I|−m*X

*n=0*

*(−1)*^{n}

*|I| − m*
*n*

( ¯*I*_{∗}^{(0)}*(c − n) − ¯I*_{∗}^{(1)}*(c − n));*

*furthermore*

(26) *I*^{∗}*(c) +*

*|I|−m*X

*n=0*

*|I| − m*
*n*

*I*¯_{∗}*(c − n) ∈ D([n*_{1}*, . . . , n** _{k}*])

*if c 6= n +*P

*s∈J**1/n*_{s}*for any n = 0, 1, . . . , |I| − m and J ⊆ ¯I with n ≡ |J|*

*(mod 2).*

*Corollary 3.4. Let |I| ≤ m and J ⊆ ¯I. Suppose that*P

*s∈J**1/n*_{s}*cannot*
*be expressed in the form n +*P

*s∈I**x*_{s}*/n*_{s}*where n ∈ {0, 1, . . . , m − |I|} and*
*x**s* *∈ R(n**s**) for s ∈ I. Put*

*J =*

*J*^{0}*⊆ ¯I :* X

*s∈J*^{0}

1

*n**s* =X

*s∈J*

1
*n**s*

*.*

*Then either |J | ≥ p([n*_{1}*, . . . , n*_{k}*]) or |J | ≡ 0 (mod 2); either |J*^{0}*| 6≡ |J|*

*(mod 2) for some J*^{0}*∈ J , or |J | can be expressed as the sum of some (not*
*necessarily distinct) prime divisors of [n*_{1}*, . . . , n*_{k}*].*

*P r o o f. Let c =*P

*s∈J**1/n*_{s}*. As ¯I*_{∗}*(c) = ¯I*_{∗}^{(0)}*(c) + ¯I*_{∗}^{(1)}*(c), and I*^{∗}*(c − n)*

*= 0 for every n = 0, 1, . . . , m − |I|, the desired results follow from Theo-*
rem 3.2(i).

*R e m a r k 2. In the case I = ∅ Corollary 3.4 was obtained by the author*
in [Su4].

*Corollary 3.5. Assume that |I| = m. Let l be the total number of ways*
*in which the rational c is expressed in the form* P

*s∈I**x**s**/n**s* *or* P

*s∈ ¯**I**δ**s**/n**s*

*where x*_{s}*∈ R(n*_{s}*) for s ∈ I and δ*_{s}*∈ {0, 1} for s ∈ ¯I. Then we have*
(27) *l ≥ p([n*1*, . . . , n**k**]) or* *l = 2*

*J ⊆ ¯I :*X

*s∈J*

1
*n*_{s}*= c*

*,*

*and l can be written as the sum of finitely many (not necessarily distinct)*
*prime divisors of n*_{1}*, . . . , n*_{k}*providing* P

*s∈J**1/n*_{s}*= c for no J ⊆ ¯I with*

*|J| ≡ 0 (mod 2).*

*P r o o f. Obviously l = I*^{∗}*(c) + ¯I*_{∗}*(c), and (22) or (25) says that ¯I**∗*^{(0)}*(c) −*
*I*¯*∗*^{(1)}*(c) = I*^{∗}*(c), i.e. l = 2 ¯I**∗*^{(0)}*(c). Therefore Theorem 3.2 yields Corollary 3.5.*

*Corollary 3.6. Let |I| ≥ m. Suppose that* P

*s∈I**m*_{s}*/n*_{s}*cannot be ex-*
*pressed in the form n +*P

*s∈J**1/n*_{s}*with n ∈ {0, 1, . . . , |I| − m} and J ⊆ ¯I,*
*where m**s**∈ R(n**s**) for each s ∈ I. Then*

(28)

*hx*_{s}*i*_{s∈I}*: x*_{s}*∈ R(n*_{s}*) for s ∈ I and* X

*s∈I*

*x*_{s}*n** _{s}* =X

*s∈I*

*m*_{s}*n*_{s}

*must be a finite sum of (not necessarily distinct) prime divisors of*
*[n*1*, . . . , n**k**].*

*P r o o f. Let c =* P

*s∈I**m*_{s}*/n*_{s}*. Note that ¯I*_{∗}*(c − n) = 0 for each n =*
*0, 1, . . . , |I| − m. By Theorem 3.2(ii), I*^{∗}*(c) belongs to D([n*_{1}*, . . . , n** _{k}*]).

*Clearly Corollary 3.6 in the case I = {1, . . . , k} gives the first half of*
result (III).

*Theorem 3.3. (i) If (n*_{s}*, n*_{t}*) | a*_{s}*− a*_{t}*for all s, t ∈ I, then*

*m−1*X

*n=0*

*I*¯_{∗}

*n +* *r*

*[n** _{s}*]

_{s∈I}

=

*J ⊆ ¯I :* X

*s∈J*

1
*n*_{s}

= *r*

*[n** _{s}*]

_{s∈I}(29)

*≥*
Q

*s∈I**n*_{s}*[n**s*]*s∈I*

*for each r = 0, 1, . . . , [n** _{s}*]

_{s∈I}*− 1.*

*(ii) Assume |I| = m, 0 ≤ θ < 1, and [n**s*]*s∈I**θ 6∈ Z or (n**i**, n**j**) - a**i**− a**j* *for*
*some i, j ∈ I. Then either*

(30)

*m−1*X

*n=0*

*I*¯_{∗}*(n + θ) =*

*J ⊆ ¯I :* X

*s∈J*

1
*n**s*

*= θ*

* ≥ p([n** ^{s}*]

_{s∈ ¯}*)*

_{I}*or*

*J ⊆ ¯I : 2 | |J| &* X

*s∈J*

1
*n*_{s}

*= θ*

=

*J ⊆ ¯I : 2 - |J| &* X

*s∈J*

1
*n*_{s}

*= θ*

*and hence*
(31)

*m−1*X

*n=0*

*I*¯*∗**(n + θ) =*

*J ⊆ ¯I :* X

*s∈J*

1
*n*_{s}

*= θ*

* ≡ 0 (mod 2);*

*moreover ,*
(32)

*m−1*X

*n=0*

*I*¯_{∗}*(n + θ) =*

*J ⊆ ¯I :* X

*s∈J*

1
*n**s*

*= θ*

* ∈ D([n** ^{s}*]

_{s∈ ¯}*)*

_{I}*if all the |J| mod 2 with J ⊆ ¯I and {*P

*s∈J**1/n*_{s}*} = θ are the same.*

*R e m a r k 3. When those n*_{s}*with s ∈ I are pairwise relatively prime,*
Theorem 3.3(i) yields the lower bound 1 in (29) while (14) gives the bound
2* ^{m−|I|}*.

*Corollary 3.7. If I 6= ∅ and (n**s**, n**t**) | a**s**− a**t* *for all s, t ∈ I, then*

(33) Y

*s∈I*

*n*_{s}*≤ 2*^{k−|I|}*,* *[n** _{s}*]

_{s∈I}*| [n*

*]*

_{s}

_{s∈ ¯}

_{I}*,*

*and*

(34) X

*s∈J*

1
*n**s*

*: J ⊆ ¯I*

*⊇*

*0,* 1

*[n**s*]*s∈I*

*, . . . ,[n** _{s}*]

_{s∈I}*− 1*

*[n*

*s*]

*s∈I*

*.*
P r o o f. (34) follows immediately from Theorem 3.3(i). SinceP

*s∈J**1/n**s*

*≡ 1/[n** _{s}*]

_{s∈I}*(mod 1) for some J ⊆ ¯I, [n*

*]*

_{s}

_{s∈I}*must divide [n*

*]*

_{s}

_{s∈ ¯}*. For the inequality in (33) we notice that*

_{I}2^{k−|I|}*≥*

*[n**s*][*s∈I**−1*
*r=0*

*J ⊆ ¯I :* X

*s∈I*

1
*n**s*

= *r*

*[n**s*]*s∈I*

=

*[n**s*X]*s∈I**−1*
*r=0*

*J ⊆ ¯I :* X

*s∈I*

1
*n*_{s}

= *r*

*[n** _{s}*]

_{s∈I}

*≥*

*[n** _{s}*X]

_{s∈I}*−1*

*r=0*

Q

*s∈I**n**s*

*[n** _{s}*]

*=Y*

_{s∈I}*s∈I*

*n*_{s}*.*

*R e m a r k 4. By checking (33) and (34) with I taken to be K = {1, . . . ,*
*m−1, k} and K ∪{k −1} in the previous example, we find that Corollary 3.7*
*is sharp. When (1) forms an exact 1-cover of Z and I ⊆ {1, . . . , k} contains*
*at least two elements, we cannot have (n**s**, n**t**) | a**s**− a**t* *for all s, t ∈ I with*
*s 6= t, and (34) fails to hold because for all J ⊆ ¯I we have*

X

*s∈J*

1

*n*_{s}*≤*X

*s∈ ¯**I*

1

*n*_{s}*= 1 −*X

*s∈I*

1

*n*_{s}*< 1 −* 1

*[n** _{s}*]

*=*

_{s∈I}*[n*

*]*

_{s}

_{s∈I}*− 1*

*[n*

*]*

_{s}

_{s∈I}*.*

*For any a, n ∈ Z with n > 0, each integer in a + nZ belongs to exactly*
*m members of (1) and hence*

*A** _{a(n)}*=

*b** _{s}*+

*n*

_{s}*(n, n*

*)Z*

_{s}

*s∈J*

*also forms an exact m-cover of Z where J = {1 ≤ s ≤ k : (n, n**s**) | a − a**s**},*
*b*_{s}*∈ Z and a + b*_{s}*n ≡ a*_{s}*(mod n*_{s}*) for s ∈ J. Instead of A = A*_{0(1)} we
*may apply our results to A** _{a(n)}* so as to get more general ones. See [Su4] for
examples of such transformations.

**4. Characterizations of exact m-covers of Z**

*Theorem 4.1. Let (1) be a system of arithmetic sequences. Let I ⊆*
*{1, . . . , k} and ¯I = {1, . . . , k} \ I. If |I| ≤ m then (1) is an exact m-cover*
*of Z if and only if*

(35) X

*J⊆ ¯**I*
P

*s∈J**1/n**s**=c*

*(−1)*^{|J|}*e*^{2πi}^{P}^{s∈J}^{a}^{s}^{/n}^{s}

=

*m−|I|*X

*n=0*

*(−1)*^{n}

*m − |I|*

*n*

X

*x**s**∈R(n**s*)*for**s∈I*
P

*s∈I**x**s**/n**s**=c−n*

*e*^{2πi}^{P}^{s∈I}^{a}^{s}^{x}^{s}^{/n}^{s}

*is valid for all rational c ≥ 0. If |I| ≥ m, then (1) forms an exact m-cover*
*of Z if and only if*

(36) X

*x**s*P*∈R(n**s*)*for**s∈I*

*s∈I**x**s**/n**s**=c*

*e*^{2πi}^{P}^{s∈I}^{a}^{s}^{x}^{s}^{/n}^{s}

=

*|I|−m*X

*n=0*

*(−1)*^{n}

*|I| − m*
*n*

X

*J⊆ ¯**I*
P

*s∈J**1/n**s**=c−n*

*(−1)*^{|J|}*e*^{2πi}^{P}^{s∈J}^{a}^{s}^{/n}^{s}

*holds for all rational c ≥ 0.*

*P r o o f. Put N = [n*1*, . . . , n**k**]. We assert that (1) forms an exact m-cover*
of Z if and only if we have the identity

(37)

Y*k*
*s=1*

*(1 − z*^{N/n}^{s}*e*^{2πia}^{s}^{/n}^{s}*) = (1 − z** ^{N}*)

^{m}*.*

*Apparently any zero of the left hand side of (37) is an N th root of unity.*

*Observe that for every integer x the number e** ^{−2πix/N}* is a zero of the left

*hand side of (37) with multiplicity m if and only if x lies in a*

_{s}*+ n*

*Z for*

_{s}*exact m of s = 1, . . . , k. So the assertion follows from Vi`ete’s theorem.*

*Now consider the case |I| ≤ m. Clearly the following identities are equiv-*
alent:

Y*k*
*s=1*

*(1 − z*^{N/n}^{s}*e*^{2πia}^{s}^{/n}^{s}*) = (1 − z** ^{N}*)

*Y*

^{m−|I|}*s∈I*

*(1 − (z*^{N/n}^{s}*e*^{2πia}^{s}^{/n}* ^{s}*)

^{n}

^{s}*),*

Y

*s∈ ¯**I*

*(1 − z*^{N/n}^{s}*e*^{2πia}^{s}^{/n}^{s}*) = (1 − z** ^{N}*)

*Y*

^{m−|I|}*s∈I*
*n*X_{s}*−1*
*m** _{s}*=0

*z*^{m}^{s}^{N/n}^{s}*e*^{2πim}^{s}^{a}^{s}^{/n}^{s}*,*
X

*J⊆ ¯**I*

*(−1)*^{|J|}*z*^{P}^{s∈J}^{N/n}^{s}*e*^{2πi}^{P}^{s∈J}^{a}^{s}^{/n}^{s}

=

*m−|I|*X

*n=0*

*(−1)*^{n}

*m − |I|*

*n*

*z** ^{nN}* Y

*s∈I*
*n*X_{s}*−1*
*m** _{s}*=0

*z*^{m}^{s}^{N /n}^{s}*e*^{2πia}^{s}^{m}^{s}^{/n}^{s}*.*
*By the assertion the first one holds if and only if (1) forms an exact m-cover*
of Z. Since the third one is valid if and only if (35) is true for every rational
*c ≥ 0, we get the desired result.*

*For the case |I| ≥ m, that (1) forms an exact m-cover of Z is equivalent*
to any of the identities given below:

Y

*s∈ ¯**I*

*(1 − z*^{N/n}^{s}*e*^{2πia}^{s}^{/n}^{s}*) ·*Y

*s∈I*

*(1 − z*^{N/n}^{s}*e*^{2πia}^{s}^{/n}^{s}*) = (1 − z** ^{N}*)

^{m}*,*Y

*s∈ ¯**I*

*(1 − z*^{N/n}^{s}*e*^{2πia}^{s}^{/n}^{s}*) ·*Y

*s∈I*

*(1 − (z*^{N/n}^{s}*e*^{2πia}^{s}^{/n}* ^{s}*)

^{n}*)*

^{s}*= (1 − z** ^{N}*)

*Y*

^{m}*s∈I*
*n*X*s**−1*
*m**s*=0

*z*^{m}^{s}^{N /n}^{s}*e*^{2πia}^{s}^{m}^{s}^{/n}^{s}*,*

*|I|−m*X

*n=0*

*(−1)*^{n}

*|I| − m*
*n*

*z** ^{nN}* X

*J⊆ ¯**I*

*(−1)*^{|J|}*z*^{P}^{s∈J}^{N/n}^{s}*e*^{2πi}^{P}^{s∈J}^{a}^{s}^{/n}^{s}

=Y

*s∈I*
*n*X_{s}*−1*
*m**s*=0

*z*^{m}^{s}^{N /n}^{s}*e*^{2πia}^{s}^{m}^{s}^{/n}^{s}*.*

*As the last one holds if and only if (36) does for all rational c ≥ 0, we are*
done.

*R e m a r k 5. In the case I = ∅ and c ∈ {1, . . . , m}, that (35) holds for*
*any exact m-cover (1) of Z was first observed by the author in [Su2] with*
the help of the Riemann zeta function.

*The characterization of exact m-cover (1) of Z given in Theorem 4.1*
*involves a fixed subset I of {1, . . . , k}. Now we present a new one which*
*depends on all the I ⊆ {1, . . . , k} with |I| = m.*

*Theorem 4.2. Let (1) be a system of arithmetic sequences. Then (1)*
*forms an exact m-cover of Z if and only if the relation*

(38) X

*J⊆{1,...,k}\I*
*{*P

*s∈J**1/n**s**}=θ*

*(−1)*^{|J|}*e*^{2πi}^{P}^{s∈J}^{a}^{s}^{/n}^{s}

= X

*x**s**∈R(n**s*)*for**s∈I*
*{*P

*s∈I**x**s**/n**s**}=θ*

*e*^{2πi}^{P}^{s∈I}^{a}^{s}^{x}^{s}^{/n}^{s}

*holds for all θ ∈ [0, 1) and I ⊆ {1, . . . , k} with |I| = m.*

*P r o o f. Let N = [n*_{1}*, . . . , n** _{k}*] and ¯

*I = {1, . . . , k}\I for all I ⊆ {1, . . . , k}.*

*First suppose that (1) forms an m-cover of Z. Let x be any integer and I*
*a subset of {1, . . . , k} with |I| = m. By taking z = r*^{1/N}*e** ^{2πix/N}* in (37), we
get the equality

Y*k*
*s=1*

*(1 − r*^{1/n}^{s}*e*^{2πi(x+a}^{s}^{)/n}^{s}*) = (1 − r)*^{m}*for all r ≥ 0. If I = {1 ≤ s ≤ k : n*_{s}*| x + a*_{s}*}, then*

Y

*s∈ ¯**I*

*(1 − e*^{2πi(x+a}^{s}^{)/n}* ^{s}*) Y

*s∈I*
*n*X_{s}*−1*
*x**s*=0

*e*^{2πi(x+a}^{s}^{)x}^{s}^{/n}^{s}

= lim

*r→1*

Y

*s∈ ¯**I*

*(1 − r*^{1/n}^{s}*e*^{2πi(x+a}^{s}^{)/n}* ^{s}*) Y

*s∈I*

¯ lim

*r→e**2πi(x+as)/ns*

*1 − ¯r*^{n}^{s}*1 − (¯r*^{n}* ^{s}*)

^{1/n}

^{s}= lim

*r→1*

Y

*s∈ ¯**I*

*(1 − r*^{1/n}^{s}*e*^{2πi(x+a}^{s}^{)/n}^{s}*) ·*Y

*s∈I*

*1 − r*^{1/n}^{s}*1 − r*

= lim

*r→1**(1 − r)*^{−|I|}

Y*k*
*s=1*

*(1 − r*^{1/n}^{s}*e*^{2πi(x+a}^{s}^{)/n}* ^{s}*)

= lim

*r→1**(1 − r)*^{−|I|}*(1 − r)*^{m}*= 1.*