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}
}
• Published 1 February 1995
• Mathematics
• Journal of Number Theory
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…

Descriptions of the Characteristic Sequence of an Irrational

• T. Brown
• Mathematics
• 1993
Abstract Let α be a positive irrational real number. (Without loss of generality assume 0 < α < 1.) The characteristic sequence of α is f(α) =f1f2 ···, where fn = [(n + 1)α] - [nα]. We make some

On the One-Sided Boundedness of Sums of Fractional Parts ({nα+γ}−12)

Abstract Given an irrational α in [0, 1), we ask for which values of γ in [0, 1) the sums C(m, α, γ)≔ ∑ 1⩽ n ⩽ m ({nα+γ}− 1 2 ) are bounded from above or from below for all m. When the partial

On Sums of Fractional Parts {nα+γ}

Abstract We use the continued fraction expansion ofαto obtain a simple, explicit formula for the sum C m (α, γ)= ∑ 1⩽k⩽m ({kα+γ}− 1 2 ) whenαis irrational. From this we deduce a number of elementary

Embeddings of circular orbits and the distribution of fractional parts

. Let r n,α ( i,t ) be the number of points of the sequence { t } , { α + t } , { 2 α + t } ,... that fall into the semiopen interval [0 , { nα } ), where { x } is the fractional part of x , n is an

On the distribution of multiples of real numbers

We introduce new series (of the variable α) that enable to measure the irregularity of distribution of the sequence of fractional parts {nα}. A detailed analysis of the convergence and divergence of

A variant of Ostrowski numeration

A variant of the usual Ostrowski $\alpha$-numeration that codes integers (positive as well as negative) and reals of [0, 1] (instead of [--$\alpha$, 1--$alpha$[], so that for every integer n, n and {n$\ alpha$} have the same coding sequence.

On the asymptotic behavior of Sudler products along subsequences

Let α ∈ (0, 1) and irrational. We investigate the asymptotic behaviour of sequences of certain trigonometric products (Sudler products) (PN (α))N∈N with PN (α) = N ∏

GENERALIZATION OF SUMS OF FRACTION PARTS AND THEIR APPLICATIONS TO NUMBER THEORY

In the paper a new multidimensional generalization of fraction part func-tion is introduced. We obtain a formula which express the number of points from the orbit of irrational shift

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

• Acta Math
• 1909

Question 1547

• l’Intermédiaire des Mathématiciens
• 1904

• 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