Balancing families of integer sequences

Abstract

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.