On the fractional parts of an/n

Javier Cilleruelo; Angel Kumchev; Florian Luca; Juanjo Rué; Igor E. Shparlinski

doi:10.1112/blms/bds084

On the fractional parts of aⁿ/n

Javier Cilleruelo^*, Angel Kumchev, Florian Luca, Juanjo Rué, Igor E. Shparlinski

^*Corresponding author for this work

School of Computing

Research output: Contribution to journal › Article › peer-review

5 Citations (Scopus)

Abstract

We give various results about the distribution of the sequence {a ⁿ/n}_{n ≥ 1} modulo 1, where a ≥ 2 is a fixed integer. In particular, we find an explicit infinite subsequence A such that {a ⁿ/n}_n∈A is uniformly distributed modulo 1. Also we show that for any constant c > 0 and a sufficiently large N, the fractional parts of the first N terms of this sequence hit every interval J ⊆ [0, 1] of length |J| ≥ c N^-0.475.

Original language	English
Pages (from-to)	249-256
Number of pages	8
Journal	Bulletin of the London Mathematical Society
Volume	45
DOIs	https://doi.org/10.1112/blms/bds084
Publication status	Published - Apr 2013

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AU - Cilleruelo, Javier

AU - Kumchev, Angel

AU - Luca, Florian

AU - Rué, Juanjo

AU - Shparlinski, Igor E.

PY - 2013/4

Y1 - 2013/4

N2 - We give various results about the distribution of the sequence {a n/n}n ≥ 1 modulo 1, where a ≥ 2 is a fixed integer. In particular, we find an explicit infinite subsequence A such that {a n/n}n∈A is uniformly distributed modulo 1. Also we show that for any constant c > 0 and a sufficiently large N, the fractional parts of the first N terms of this sequence hit every interval J ⊆ [0, 1] of length |J| ≥ c N-0.475.

AB - We give various results about the distribution of the sequence {a n/n}n ≥ 1 modulo 1, where a ≥ 2 is a fixed integer. In particular, we find an explicit infinite subsequence A such that {a n/n}n∈A is uniformly distributed modulo 1. Also we show that for any constant c > 0 and a sufficiently large N, the fractional parts of the first N terms of this sequence hit every interval J ⊆ [0, 1] of length |J| ≥ c N-0.475.

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EP - 256

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ER -

On the fractional parts of aⁿ/n

Abstract

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