Comparing Families of Thin Sets

Abstract

Recently the first author investigated a generalization of families of trigonometric thin sets replacing the sine function by a continuous function. In this paper we shall partially solve the problem of the relationship between such families obtained from different functions. In several cases we present conditions for equality of such a family with corresponding trigonometric family. Moreover we show that every basis of any of the families of $B_0$--sets, $N_0$--sets or $A$--sets has cardinality at least that of the continuum.