<abstract xmlns="http://www.w3.org/1999/xhtml"><p>The paper gives conditions for a sequence of fractional parts of real numbers <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2016-0008_eq_001.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mo>(</mo><mrow><mo>{</mo><msub><mi>a</mi><mi>n</mi></msub><mi>x</mi><mo>}</mo></mrow><mo>)</mo></mrow><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mi>∞</mi></msubsup></math><tex-math>$\left( {\{ a_n x\} } \right)_{n = 1}^\infty $</tex-math></alternatives></inline-formula> to satisfy a pair correlation estimate. Here <italic>x</italic> is a fixed nonzero real number and <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2016-0008_eq_002.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mo>(</mo><mrow><msub><mi>a</mi><mi>n</mi></msub></mrow><mo>)</mo></mrow><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mi>∞</mi></msubsup></math><tex-math>$\left( {a_n } \right)_{n = 1}^\infty $</tex-math></alternatives></inline-formula> is a random walk on the integers.</p></abstract>

The paper gives conditions for a sequence of fractional parts of real numbers ({anx})n=1∞$\left( {\{ a_n x\} } \right)_{n = 1}^\infty $ to satisfy a pair correlation estimate. Here x is a fixed nonzero real number and (an)n=1∞$\left( {a_n } \right)_{n = 1}^\infty $ is a random walk on the integers.

Pair Correlations and Random Walks on the Integers

Department of Mathematical Sciences, University of Liverpool, Mathematical Sciences Building, Liverpool L69 7ZL,

Uniform distribution theory

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

{"article-title":"Pair Correlations and Random Walks on the Integers"}

The paper gives conditions for a sequence of fractional parts of real numbers ({anx})n=1∞$\left( {\{ a_n x\} } \right)_{n = 1}^\infty $ to satisfy a pair...

Pair Correlations and Random Walks on the Integers

Published Online: Jan 13, 2017

Page range: 159 - 164

Received: Mar 10, 2015

Accepted: Mar 22, 2016

DOI: https://doi.org/10.1515/udt-2016-0008

Keywords
pair correlation, random walks

© 2016 Radhakrishnan Nair et al., published by De Gruyter Open

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

Pair Correlations and Random Walks on the Integers

Published Online: Jan 13, 2017

Page range: 159 - 164

Received: Mar 10, 2015

Accepted: Mar 22, 2016

DOI: https://doi.org/10.1515/udt-2016-0008

Keywordspair correlation, random walks

© 2016 Radhakrishnan Nair et al., published by De Gruyter Open

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

Keywords
pair correlation, random walks