## The structure of complementary sets of integers: a 3-shift theorem

Venue: | INTERNAT. J. PURE AND APPL. MATH |

Citations: | 8 - 6 self |

### BibTeX

@ARTICLE{Fraenkel_thestructure,

author = {Aviezri S. Fraenkel and Dalia Krieger},

title = {The structure of complementary sets of integers: a 3-shift theorem},

journal = {INTERNAT. J. PURE AND APPL. MATH},

year = {},

volume = {10},

pages = {1--49}

}

### OpenURL

### Abstract

Let 0 < α < β be two irrational numbers satisfying 1/α +1/β = 1. Then the sequences a ′ n = ⌊nα⌋, b ′ n = ⌊nβ⌋, n ≥ 1, are complementary: 1 ≤ i < n}, n ≥ 1 over Z≥1, thus a ′ n satisfies: a ′ n = mex1{a ′ i, b ′ i (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.