<abstract xmlns="http://www.w3.org/1999/xhtml"><p>In 1923 A. Khinchin asked if given any B ⊆ [0, 1) of positive Lebesgue measure, we have <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/Untitled-2.jpg"></inline-graphic></alternatives></inline-formula> #{n : 1 ≤ n ≤ N : {nx} ∈ 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. </p></abstract>

In 1923 A. Khinchin asked if given any B ⊆ [0, 1) of positive Lebesgue measure, we have  #{n : 1 ≤ n ≤ N : {nx} ∈ 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.

Problems in Strong Uniform Distribution

Mathematical Sciences The University of Liverpool Peach Street Liverpool L69 7ZL United Kingdom

Tatra Mountains Mathematical Publications

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

{"article-title":"Problems in Strong Uniform Distribution"}

In 1923 A. Khinchin asked if given any B ⊆ [0, 1) of positive Lebesgue measure, we have  #{n : 1 ≤ n ≤ N : {nx} ∈ B} → |B| for almost all x with respect to...

Problems in Strong Uniform Distribution

Published Online: Mar 11, 2015

Page range: 51 - 64

Received: Oct 06, 2014

DOI: https://doi.org/10.2478/tmmp-2014-0018

© 2015

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.