On the Function w(x)=|{1≤sk : xa s (mod n s )}|

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 ≤ sk : xa 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 kl+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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Correspondence to Zhi-Wei Sun.

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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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Sun, ZW. On the Function w(x)=|{1≤sk : xa 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