A Model for Pairs of Beatty Sequences

Beatty sequences [nα1 + β1]n∈Z and [mα2 + β2]m∈Z are recorded by two athletes running in opposite directions in a round stadium. This approach suggests a nice interpretation for well known partitioning criteria: such sequences (eventually) partition the integers essentially when the athletes have the same starting point. A remarkable observation due to S. Beatty says that if w is any positive irrational number, then the sequences 1 + w, 2(1 + w), 3(1 + w), ... 1 + 1 w , 2(1 + 1 w ), 3(1 + 1 w ), ... contain one and only one number between each pair of consecutive positive integers. Denoting α1 := 1 + w, α2 := 1 + 1 w , the corresponding sequences of (floor) integer parts S(α1) := [nα1]n∈N, S(α2) := [mα2]m∈N are called Beatty sequences with moduli α1, α2 respectively. Note that the sum of α1 and α2 is equal to their product, i.e. they satisfy (0.1) 1 α1 + 1 α2 = 1. Beatty’s result can thus be reformulated as follows: Theorem 1. (Beatty, see [1]) Let α1, α2 be two positive irrational numbers satisfying 1 α1 + 1 α2 = 1, then the Beatty sequences S(α1), S(α2) partition N. The converse of Theorem 1 is also valid: since the density of a Beatty sequence [nα]n∈N in N is equal to 1 α , then S(α1), S(α2) partition N only if α1 and α2 satisfy (0.1) (and hence there exists a positive number w such that α1 = 1 + w and α2 = 1 + 1 w ). In 1957 Th. Skolem generalized the above theorem to non-homogeneous Beatty sequences, i.e. double infinite sequences of the form S(α, β) := [nα+ β]n∈Z, α ∈ R , β ∈ R. When α1 and α2 are irrational, the question is when S(α1, β1), S(α2, β2) eventually partition Z, that is any sufficiently large (and any sufficiently small) integer belongs exactly to one of the sequences. Theorem 2. (Skolem [11], see also [3, 5]) Let α1, α2 be two positive irrational numbers satisfying 1 α1 + 1 α2 = 1, and let β1, β2 be real numbers. Then S(α1, β1), S(α2, β2) Date: March 29, 2011.