Balancing families of integer sequences


In this paper we prove the following theorem: Given a sequenceA 1,A 2, ...;A k ={a (k)1 <a (k)2 <...} of infinite sets of positive integers, there exists a suitable functiong(n)=± 1 for which

$$\mathop {\max }\limits_m \left| {\sum\limits_{i = 1}^m {g(a_i^{(k)} )} } \right|< k^{(1 + \varepsilon )\log k/2} if k \geqq k_0 (\varepsilon ).$$

Some generalizations are also considered.

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Beck, J. Balancing families of integer sequences. Combinatorica 1, 209–216 (1981).

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AMS subject classification (1980)

  • 10 L 20
  • 05 C 65
  • 05 C 55