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In 2018, some mathematicians proved that any non-negative integer subset with a positive limit density (for some eps>0 and all large enough n, the subset contai
by tromp 4y ago
In 2018, some mathematicians proved that any non-negative integer subset with a positive limit density (for some eps>0 and all large enough n, the subset contains at least eps*n numbers below n) contains a sumset, i.e. a set C that can be written as the sum of two infinite sets C = A + B (the set of a+b where a in A and b in B). More recently, they extended this to sums of more than two infinite sets.
- _a_a_a_ 4y agoI'm afraid that does not illuminate at all.
- deleted 4y ago[deleted]
- brookst 4y agoCan someone explain this comment for the sub-cretins in the room?
- Someone 4y agoLet’s start with finite sets, and in reverse: given the two sets A and B, define set C = {a + b | a ∈ a, b ∈ B} For example if A = {1, 3, 7} and B = {1, 4, 8} , we have 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} That’s easy. Now, if you didn’t know that, but were given set C, can you find two sets A and B of size 3 that would produce C under that logic? 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.
- brookst 4y agoThank you!