Approximation of ⌊nα + s⌋ and the Zero of {nα + s}

AbstractLet α and β be positive real numbers and s a real number satisfying 0 ≤ s < 1. Let ⌊x⌋ denote the greatest integer ≤ x, and {x} = x − ⌊x⌋. Define Ψ (α, β; s) to be the least positive integer n such that ⌊nα + s⌋ ≠ ⌊nβ + s⌋. When s = 0, a simple explicit formula for Ψ is given, and otherwise more complicated formulas are obtained. When α is irrational, a formula for finding the least n ∈ Z+ such that {nα + s} = 0 is presented. A natural characterization of the approximation properties of intermediate convergents (including convergents) without reference to the apparatus of continued fractions is given. A new characterization of the sequence ⌊nα⌋ for n ≥ 1 is found, and ⌊nα + s⌋ for n ≥ 1 is also characterized in a different way