Inclusions for Classes of Lacunary Sets

Abstract

A sequence, a1 < a2 < a3 < …, of positive integers is called lacunary if the difference sequence dn = an+l — an tends to infinity as n → ∞.In several recent papers we have made use of these sequences in analysis and combinatorics. In [6] we show that the class of all sets which are either finite or the range of a lacunary sequence is “full” in the sense that if (tk) is a real sequence and for each then (tk) is an l1 sequence, that is,In [3] the class of all finite unions of sets of is shown to consist of exactly those sets of integers, A, whose characteristic sequence, χA, is in the well known summability space bs + c0. More recently, in [1], we study lacunary sequences in connection with the conjecture of P.