On maximal (k, b)-linear-free sets of integers and its spectrum

Let k and b be integers with k > 1. A set A of integers is called (k, b)linear-free if x E A implies kx + b ¢ A. Such a set A is maximal in [I,n] = {1,2, ... ,n} if AU{t} is not (k,b)-linear-free for any t in [1,nJ\A. Let M = M(n, k, b) be the set of all maximal (k, b)-linear-free subsets of [1, n] and define f(n, k, b) = max{IAI : A E M} and g(n, k, b) = min{IAI : A EM}. In this paper a new method for constructing maximal (k, b)-linear-free subsets of [I,n] is given and formulae for f(n, k, b) and g(n, k, b) are obtained. Also, we investigate the spectrum of maximal (k, b)-linear-free subsets of [1, n], and prove that there is a maximal (k, b)linear-free subset of [1, n] with x elements for any integer x between the minimum and maximum possible orders.