Claims

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1982_01_01_klarner: A 1982 theorem that affine maps x to a_i x + b_i with positive integer a_i generate a non-free semigroup when the reciprocals of the a_i sum to more than one; two distinct words of equal length agree, so some A_k repeats.

2021_05_19_kolpakov_talambutsa: Theorem 3 of a 2022 paper: affine maps with positive integer multipliers and rational shifts generate a non-free semigroup when the reciprocals of the multipliers sum to more than one; the problem is a special case.

2025_12_01_barreto: An elementary proof posted in the site's thread: the reciprocal sum over A_k grows geometrically when the entries are distinct but is bounded by a harmonic number of order k; accepted by the site as Klarner's proof refound.

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