Syndetic Partitions of the Set of Positive Integers Based on Binary Sequences

The paper rigorously proves that the set of positive integers can be partitioned into two syndetic sets such that each set intersects every infinite arithmetic progression. This is achieved by describing a binary sequence generation algorithm, from which two sets are derived based on the indices of 1's and 0's. It is demonstrated that both sets are syndetic and that every infinite arithmetic progression intersects both sets.