Punctual dimension and the punctual degrees of discrete linear orderings

In the first part of this paper, we will prove a new sufficient condition for a structure to have infinite punctual dimension and then apply it to show several new connections between the punctual dimension of a structure and its punctual degrees. In the second part of this paper, we contribute toward a classification of the density of punctual degrees for linear orders by proving that the punctual degrees of Z ⋅ n , for n > 1 , are dense.