Infinite patterns appear in numbers described as moving systems

  C = {a + b | a ∈ a, b ∈ B}

  A = {1, 3, 7}

  B = {1, 4, 8}

  C = {1 + 1, 1 + 4, 1 + 8,
       3 + 1, 3 + 4, 3 + 8,
       7 + 1, 7 + 4, 7 + 8}
    = {2, 5, 9, 4, 7, 11, 8, 11, 15}
    = {2, 4, 5, 7, 8, 9, 11, 15}

An easy (but cumbersome) way to answer that is: produce all sets of 3 positive integers less than the largest number in C (15 in the example), compute the respective C’s and check whether that’s equal to the targeted C.

This problem is about targeting infinite sets of integers for A, B, and C, with an extra twist: for what infinite sets of integers C can you find matching infinitely sized sets A and B that both are subsets of C?

Since all of these sets are infinite, the easy but cumbersome method won’t work; neither set has a largest element. In addition, the result isn’t about a specific set of integers.

all I can tell you (because it's all I know) is that "the structure" is preserved by transformations.

and what the subject focuses on and studies is the transformations, more than 'the structures'.

so I suppose they don't precisely and explicitly know what 'the structure' is; but the incredible thing is how this does NOT matter.

as far as I've figured out so far, the point is that it gets preserved across transformations, and that they can inter-relate it across different mathematical objects.