Sums of Fractional Parts of Integer Multiples of an Irrational
@article{Brown1995SumsOF, title={Sums of Fractional Parts of Integer Multiples of an Irrational}, author={Tom C. Brown and Peter J.-S. Shiue}, journal={Journal of Number Theory}, year={1995}, volume={50}, pages={181-192} }
Let α be a positive irrational real number, and let Cα(n) = ∑1 ≤ k ≤ n ({kα} − 12), n ≥ 1, where {x} denotes the fractional part of x. We give an explicit formula for Cα(n) in terms of the simple continued fraction for α, and use this formula to give simple proofs of several results of A. Ostrowski, G. H. Hardy and J.E. Littlewood, and V. T. Sos. We also show that there exist positive constants dA such that if α = [a0, a1, a2, ...] and (1/t) ∑1 ≤ j ≤ taj ≤ A holds for infinitely many t, then C…
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References
SHOWING 1-10 OF 10 REFERENCES
Determination of $[n heta ]$ by its sequence of differences
- Mathematics
- 1978
Abstract For any real number θ let where [x] denotes the greatest integer not exceeding x. A method is given for computing fθ from its first few terms. A similar method is given for computing the…
Some problems of Diophantine approximation: The lattice-points of a right-angled triangle. (Second memoir.)
- Mathematics
- 1922
Some Problems of Diophantine Approximation: The Lattice-Points of a Right-Angled Triangle
- Mathematics
- 1922
Sós , On the theory of diophantine approximations i ( on a problem of a . ostrowski )
- Acta Math
- 1909
Question 1547
- l’Intermédiaire des Mathématiciens
- 1904
Real numbers with bounded partial quotients : a survey , Enseign
- 1922
Pewne twierdzenie tyczace sie liczb niewymiernych. – un théorème sur les nombres irrationnels
- Bull. Internat. Acad. Polon. Sci. Lett. Cl. Sci. Math. Naturelles Sér. A (Cracovie)
- 1909
Real numbers with bounded partial quotients: a survey
- Enseign. Math
- 1992