Pair correlations, beta distribution and hypergeometric functions

The <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N"> <mml:semantics> <mml:mi>N</mml:mi> <mml:annotation encoding="application/x-tex">N</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-point pair correlation statistic has been studied in detail for uniformly distributed sequences in the unit interval. Generically, the pair correlation statistic of uniformly distributed random sequences has so-called Poissonian pair correlation in its limit. For any (arbitrary) sequence in the unit interval, the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N"> <mml:semantics> <mml:mi>N</mml:mi> <mml:annotation encoding="application/x-tex">N</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-point pair correlation static can by definition be bounded by <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N minus 1"> <mml:semantics> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">N-1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In this paper, it is shown that any intermediate order of growth <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N Superscript alpha"> <mml:semantics> <mml:msup> <mml:mi>N</mml:mi> <mml:mi>α</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">N^\alpha</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0 greater-than alpha greater-than 1"> <mml:semantics> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>&gt;</mml:mo> <mml:mi>α</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">0 &gt; \alpha &gt; 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> can be realized by beta distributed random variables.