Part 3. Mathematical models

About the dimension difference of periodic subsets of the natural series

P. S. Dergach^a, L. R. Bulgakov<sup>b

a</sup> Lomonosov Moscow State University
^b National Research Nuclear University MEPhI

Abstract: This work is devoted to describing the change in the dimension of periodic subsets of the natural series with seemingly insignificant operations like removal/addition to the set of one number. The case is investigated when the dimension of the initial set is equal to 1 or 2. By the dimension of the set is meant the minimum number of disjoint arithmetic progressions that give this set in the union. For sets of dimension 2 the result is obtained only in cases of pairs of general position progressions. In this paper we give the results on how the dimension changes depending on whether the number $x$ is deleted/added to.

Keywords: arithmetic progression, natural series, progressive set.