The Structure of Complementary Sets of Integers: a 3-shift Theorem

Let 0 < α < β be two irrational numbers satisfying 1/α+1/β = 1. Then the sequences a′n = bnαc, b′n = bnβc, n ≥ 1, are complementary over Z≥1, thus a′n satisfies: a′n = mex1{ai, bi : 1 ≤ i < n}, n ≥ 1 (mex1(S), the smallest positive integer not in the set S). Suppose that c = β − α is an integer. Then b′n = a′n + cn for all n ≥ 1. We define the following generalization of sequences a′n, b′n: Let c, n0 ∈ Z≥1, and let X ⊂ Z≥1 be an arbitrary finite set. Let an = mex1(X∪{ai, bi : 1 ≤ i < n}), bn = an+cn, n ≥ n0. Let sn = a′n−an. We show that no matter how we pick c, n0 and X, from some point on the shift sequence sn assumes either one constant value or three successive values; and if the second case holds, it assumes these values in a very distinct fractal-like pattern, which we describe. This work was motivated by a generalization of Wythoff’s game to N ≥ 3 piles. AMS Subj. Classification: 05A17, 91A05