<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

Publié en ligne: 13 janv. 2017

Pages: 159 - 164

Reçu: 10 mars 2015

Accepté: 22 mars 2016

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

Mots clés
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

Publié en ligne: 13 janv. 2017

Pages: 159 - 164

Reçu: 10 mars 2015

Accepté: 22 mars 2016

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

Mots cléspair 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.

Mots clés
pair correlation, random walks