Product-intersection sets of asymptotically decreasing sequences
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For subsets A_q of a semigroup, the product intersection set consists of positive integers h for which (∩_q A_q)^h=∩_q(A_q^h), where B^h is the set of h-fold products of elements of B. Which sets of exponents can arise?
The question includes arbitrary index sets Q and decreasing sequences that are not eventually constant. The proposed classification for |Q|≥2, and separately for those decreasing sequences, is every subset of the positive integers containing 1.
Why it matters and what counts as progress
Why it matters: A complete answer identifies the possible intersection patterns, beyond known closure under intersections.
What counts as progress: Constructions realizing every permitted set and necessity proofs, treating |Q|=0, |Q|=1 and |Q|≥2 separately.
How the problem developed
Melvyn B. Nathanson Asked which subsets of the positive integers can occur as product-intersection sets, including for asymptotically strictly decreasing sequences of sets.
- 20 Apr 2026 · Classification [2]
Wouter van Doorn, Pietro Monticone and Quanyu Tang Gave a complete classification for arbitrary families and asymptotically strictly decreasing sequences; when at least two sets are indexed, every subset containing 1 occurs. The paper reports autonomous discovery and Lean verification by Aristotle.
AI contribution
Global Product Intersection Sets in Semigroups
For asymptotically strictly decreasing sequences, and for every index set Q with at least two members, the possible global product-intersection sets are exactly the subsets of positive integers containing 1. The paper also specifies the Q=0 and Q=1 cases.
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