INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 5(2) (2005), #A03 DIFFERENCE DENSITY AND APERIODIC SUM-FREE SETS

Cameron has introduced a natural one-to-one correspondence between infinite binary se-quences and sets of positive integers with the property that no two elements add up to a third. He observed that, if a sum-free set is ultimately periodic, so is the corresponding binary sequence, and asked if the converse also holds. We introduce the concept of difference density and show how this can be used to test specific sets. These tests produce further evidence of a positive nature that certain sets are, in fact, not ultimately periodic. 1