The answer seems to be **yes** by 

[*Parametric Presburger arithmetic: logic, combinatorics, and quasi-polynomial behavior*][1]

by Kevin Woods, John Goodrick, and Tristram Bogart.  DOI:10.19086/da.1254v2


  [1]: http://discreteanalysisjournal.com/article/1254-parametric-presburger-arithmetic-logic-combinatorics-and-quasi-polynomial-behavior