Primitive Sets with Small Gaps

A primitive set is a set S of positive integers such that no pair of distinct elements a,b\in S satisfies a | b.

A square-primitive set is a set of positive integers S such that no pair of distinct elements a, b \in S has ratio a/b equal to an square integer.

In the following writeup, I show how to construct square-primitive sets with very short gaps O_{\epsilon} ((\log N)^2 (\log \log N)^{\epsilon}). This relies on dividing the poset \mathbb{N} ordered by divisibility into antichains A_m consisting of all numbers in \mathbb{N} with exactly m (not necessarily distinct) prime factors, and then picking one randomly.

Square Primitive Sets With Small Gaps

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