Approximation of left floornα + sright floor and the Zero of {nα + s} Original Research Article

Let α and β be positive real numbers and s a real number satisfying 0 ≤ s < 1. Let left floorxright floor denote the greatest integer ≤ x, and {x} = x − left floorxright floor. Define Ψ (α, β; s) to be the least positive integer n such that left floornα + sright floor ≠ left floornβ + sright floor. 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 set membership, variant image+ 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 left floornαright floor for n ≥ 1 is found, and left floornα + sright floor for n ≥ 1 is also characterized in a different way.