On the Generating Function of the Integer Part: [nα + γ] Original Research Article

We show that [formula]. Here pn and qn are the numerators and denominators of the convergents of the continued fraction expansion of α and t**n and s**n are particular algorithmically generated sequences of best approximates for the non-homogeneous diophantine approximation problem of minimizing nα + γ − m. This generalizes results of Böhmer and Mahler, who considered the special case where γ = 0. This representation allows us to easily derive various transcendence results. For example, ∑∞n=1 [ne + image]/2n is a Liouville number. Indeed the first series is Liouville for rational z, wset membership, variant [−1, 1] with zw ≠ 1 provided α has unbounded continued fraction expansion. A second application, which generalizes a theorem originally due to Lord Raleigh, is to give a new proof of a theorem of Fraenkel, namely [nα + γ]∞n=1 and [nα′ + γ′]∞n=1 partition the non-negative integers if and only if 1/α + 1/α′ = 1 and γ/α + γ′/α′ = 0 (provided some sign and integer independence conditions are placed on α, β, γ, γ′). The analysis which leads to the results is quite delicate and rests heavily on a functional equation for G. For this a natural generalization of the simple continued fraction to Kronecker′s forms nα + γ − m is required.