Gap Problems for Integer Part and Fractional Part Sequences

Abstract

Our main concern is with Beatty sequences, i.e., sequences of the form {⌊ nα + γ⌋: n = 0, 1, ...}, where α, γ are real numbers (α ≥ 1 is called the modulus of the sequence). We look at the intersection of two Beatty sequences, and ask how many distinct gaps there are (a gap is the difference between two consecutive elements of the intersection). This problem turns out to be closely connected to two gap problems involving fractional part sequences of the form {{ nα + γ}: n = 0, 1, ...}, introduced by Steinhaus and Slater, respectively. Our main results for the intersection problem and their connections to the other two problems are as follows. When one of the Beatty sequences is arithmetical (i.e., its modulus is an integer), we apply a three-gap theorem of Slater to show that there are at most three gaps; if there are three gaps, one of them is the sum of the other two. This result was obtained independently by Wolff and Pitman. When at least one of the Beatty sequences has a rational modulus, we ask for an upper bound on the number of gaps as a function of the denominator q of the rational modulus (or the smaller of the two denominators, if both moduli are rational). For q ≥ 2, we show that the best upper bound is q + 3. The upper bound follows from a recent result of Geelen and Simpson for a two-dimensional version of the Steinhaus problem, motivated by the current work. Finally, we prove that the intersection of two arbitrary Beatty sequences has finitely many gaps, by establishing a corresponding finiteness result for a two-dimensional version of the Slater problem.