- Title
- On the generating function of the integer part: [nα + γ]
- Creator
- Borwein, J. M.; Borwein, P. B.
- Relation
- Journal of Number Theory Vol. 43, Issue 3, p. 293-318
- Publisher Link
- http://dx.doi.org/10.1006/jnth.1993.1023
- Publisher
- Academic Press
- Resource Type
- journal article
- Date
- 1993
- Description
- We show that [formula could not be replicated]. 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 +1/2 ]/2n is a Liouville number. Indeed the first series is Liouville for rational z, w∈ [−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.
- Subject
- integer; Fraenkel; Liouville; Kronecker; fraction expansion
- Identifier
- http://hdl.handle.net/1959.13/1042533
- Identifier
- uon:14077
- Identifier
- ISSN:0022-314X
- Language
- eng
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