How to check whether an infinite set is closed under addition with a computer code?

Question

Given k positive integer numbers a₁ < a₂ < a₃ < ... < a_k, and all integers greater than a_k, we want to check whether the set A = {a_i : i ∈ [1,k]} ∪ {n : n > a_k, n ∈ ℕ } = {a₁, a₂, a₃, ... , a_k, a_k+1, a_k+2, ...} is closed under addition. This means that:

∑_{1 ≤ i ≤ k} a_i*b_i ∈ A, for any non-negative integers b_i.

For example, {2,4,6,7,8,....} is closed under addition.

Is there any easy way to do this? Any functions that we could use in Mathematica or Matlab?

Answer 1

If the discontinuous portion less than k of the sets are not large I believe you can approach it directly like this:

a = {2, 4, 6};
Tr /@ Subsets[a, {2}];
TakeWhile[%, # < Last@a &];
Complement[%, a] === {}

Answer 2

A banal observation: any sum having an operand >= a_k is a member of A, so you only need to be concerned with sums where both operands are from the set B = {a_1 .. a_(k-1)}. Naive solution in pseudocode:

for i from k-1 down to 1
  for j from i down to 1
    if (a_i + a_j < a_k) and (a_i + a_j is not in B) then
      return false
return true