For a finite system
\( A = {\left\{ {a_{s} + n_{s} \mathbb{Z}} \right\}}^{k}_{{s = 1}} \) of arithmetic sequences
the covering function is *w*(*x*)
= |{1 ≤ *s* ≤
*k* : *x* ≡ *a*
_{
s
} (mod
*n*
_{
s
})}|. Using equalities
involving roots of unity we characterize those systems with a
fixed covering function *w*(*x*). From the characterization we reveal
some connections between a period *n*
_{0} of
*w*(*x*) and the moduli
*n*
_{1}, .
. . , *n*
_{
k
} in such a system
*A*. Here are three central
results: (a) For each *r*=0,1,
. . .,*n*
_{
k
}/(*n*
_{0},*n*
_{
k
})−1 there exists a
*J*
__c__{1, . . . ,
*k*−1} such that
\( {\sum\nolimits_{s \in J} {1/n_{s} = r/n_{k} } } \). (b) If
*n*
_{1}
≤···≤*n*
_{
k−l
} <*n*
_{
k−l+1} =···=*n*
_{
k
} (0 <
*l* <
*k*), then for any positive
integer *r* <
*n*
_{
k
}/*n*
_{
k−l
} with
*r* ≢ 0 (mod
*n*
_{
k
}/(*n*
_{0},*n*
_{
k
})), the binomial
coefficient
\( {\left( {\begin{array}{*{20}c} {l} \\ {r} \\ \end{array} } \right)} \) can be written as the
sum of some (not necessarily distinct) prime divisors of
*n*
_{
k
}. (c)
max_{(x∈ℤ}
*w*(*x*)
can be written in the form
\( {\sum\nolimits_{{\left( {s = 1} \right)}}^k {m_{s} /n_{s} } } \) where
*m*
_{1}, .
. .,*m*
_{
k
} are positive
integers.

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### Affiliations

### Corresponding author

## Additional information

The research is supported by the Teaching and Research Award Fund for Outstanding Young Teachers in Higher Education Institutions of MOE, and the National Natural Science Foundation of P. R. China.

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### Cite this article

Sun, ZW. On the Function
*w*(*x*)=|{1≤*s*≤*k*
: *x*≡*a*
_{
s
} (mod
*n*
_{
s
})}|.
*Combinatorica* **23, **681–691 (2003). https://doi.org/10.1007/s00493-003-0041-0

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*Mathematics Subject
Classification (2000):*

- 11B25
- 05A15
- 11A07
- 11A25
- 11B75
- 11D68