Problems in strong uniform distribution

Kwo Chan; Radhakrishnan Nair

doi:10.2478/tmmp-2014-0018

Problems in strong uniform distribution

Kwo Chan
, Radhakrishnan Nair

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

Abstract

In 1923 A. Khinchin asked if given any B ? [0, 1) of positive Lebesgue measure, we have #{n : 1 ≤ n ≤ N : {nx} ε B} → |B| for almost all x with respect to Lebesgue measure. Here {y} denotes the fractional part of the real number y and |A| denotes the Lebesgue measure of the set A in [0, 1). In 1970 J. Marstrand showed the answer is no. In this paper the authors survey contributions to this subject since then.

Original language	English
Pages (from-to)	51-64
Number of pages	14
Journal	Tatra Mountains Mathematical Publications
Volume	59
Issue number	1
DOIs	https://doi.org/10.2478/tmmp-2014-0018
Publication status	Published - 1 Jun 2014
Externally published	Yes

Keywords

ergodic averages
strong uniform distribution

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10.2478/tmmp-2014-0018

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abstract = "In 1923 A. Khinchin asked if given any B ? [0, 1) of positive Lebesgue measure, we have \#\{n : 1 ≤ n ≤ N : \{nx\} ε B\} → |B| for almost all x with respect to Lebesgue measure. Here \{y\} denotes the fractional part of the real number y and |A| denotes the Lebesgue measure of the set A in [0, 1). In 1970 J. Marstrand showed the answer is no. In this paper the authors survey contributions to this subject since then.",

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AU - Nair, Radhakrishnan

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AB - In 1923 A. Khinchin asked if given any B ? [0, 1) of positive Lebesgue measure, we have #{n : 1 ≤ n ≤ N : {nx} ε B} → |B| for almost all x with respect to Lebesgue measure. Here {y} denotes the fractional part of the real number y and |A| denotes the Lebesgue measure of the set A in [0, 1). In 1970 J. Marstrand showed the answer is no. In this paper the authors survey contributions to this subject since then.

KW - ergodic averages

KW - strong uniform distribution

UR - https://www.scopus.com/pages/publications/84924860132

U2 - 10.2478/tmmp-2014-0018

DO - 10.2478/tmmp-2014-0018

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VL - 59

SP - 51

EP - 64

JO - Tatra Mountains Mathematical Publications

JF - Tatra Mountains Mathematical Publications

IS - 1

ER -

Problems in strong uniform distribution

Abstract

Keywords

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