<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

Publicado en línea: 13 ene 2017

Páginas: 159 - 164

Recibido: 10 mar 2015

Aceptado: 22 mar 2016

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

Palabras clave
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

Publicado en línea: 13 ene 2017

Páginas: 159 - 164

Recibido: 10 mar 2015

Aceptado: 22 mar 2016

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

Palabras clavepair 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.

Palabras clave
pair correlation, random walks