About the maximum coverage of positive integers with some kind of restrictions

In this thesis it is necessary to find the maximum length of the beginning of natural set, that can be covered by the union of arithmetic progressions without covering this way all natural set. There are also some kind of restrictions on beginning and step of these progressions, and on their total number. Depending on what type of restrictions take place, we have a class of various tasks. Some of them were solved in this paper. The most interesting cases are types of restrictions like “beginning+step"and "quantity”.