# Pair Correlations and Random Walks on the Integers

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