Pair correlation of values of rational functions (mod \(p\))
Let \(x_n\), \(n=1,2,\dots\), be a sequence of points in the unit interval \([0,1)\), and for intervals \(I=[a,b]\subset\mathbb R\) denote \[ I/N=[a/N,b/N],\qquad I/N\pm 1=[(a/N)\pm 1,(b/N)\pm 1] \] and \(|I|=b-a\). The authors study distribution properties of differences \(x_i-x_j\), \(1\leq i\neq j\leq N\), with respect to intervals of the type \(I/N\) whose lengths depend on \(N\). Precisely, the authors call the quantity \[ R_2=N^{-1}\#\{1\leq i\neq j\leq N;x_i-x_j\in I/N\cup(I/N+1)\cup(I/N-1)\} \] pair correlation (nontraditionally) of \(x_n\) and this pair correlation is Poissonian if \(R_2\to|I|\), as \(N\to\infty\). Note that, for \(I=[-s,s]\), we have \(R_2=N^{-1}\#\{1\leq i\neq j\leq N\); \(\|x_i-x_j\|\in[0,s/N]\}\), where \(\|x\|=\min(\{x\},1-\{x\})\) and \(\{x\}\) is the fractional part of \(x\). \textit{P. Sarnak} and \textit{Z. Rudnick} [Commun. Math. Phys. 194, 61-70 (1998; Zbl 0919.11052)] show that the pair correlation of \(\alpha n^d\bmod 1\), for integral \(d\geq 2\), is Poissonian for almost all \(\alpha\). For individual \(\alpha\) the authors prove the main result (Theorem 1.4): For any irrational \(\alpha\) and any \(\sigma\geq 5\), there exist two increasing sequences of positive sequences \(M_j,N_j\) with \((\log M_j)/(\log N_j)\to\sigma\) such that the pair correlation of \(\alpha n^2\bmod 1\) for \(M_j<n\leq M_j+N_j\) is Poissonian as \(j\to\infty\). Replacing \(\alpha n^2\) by \((k/q)n^2\) for a proof, they investigate sequences \(r(n)\pmod p\), \(m<n\leq m+N\), where \(r(x)=f(x)/g(x)\) is a rational function with integer coefficients and \(p\) is a large prime. For the related pair correlation \[ \begin{multlined} R_2=N^{-1}\#\{m<x\neq y\leq m+N;g(x),g(y)\neq 0\\ \pmod p, r(x)-r(y)=h\pmod p\text{ for some }h\in\mathbb Z,\;h/p\in I/N\}\end{multlined} \] they prove that \(|R_2-|I||\leq sp^{-\delta}\) for \(0<\delta<1/16\), any interval \(I\subset[-s,s]\), almost all \(m\in\{1,2,\dots, M\}\), where \(M\) and \(N\) grow in a prescribed way together with the prime \(p\) to infinity. In this proof the basic lemma (Proposition 2.2) states: Denote by \(L\) the number of integral \(6\)-tuples \((x_1,y_1,x_2,y_2,t,v)\) satisfying \(1\leq x_1,y_1,x_2,y_2\leq N\), \(|t_1|\in[A,2A]\), \(|v|\in[B,2B]\), \(r(x_j)\neq r(y_j)\), \(g(x_j)\neq 0\), \(g(y_j)\neq 0\), \(v(r(x_2)-r(y_2))=t(r(x_1)-r(y_1))\) all of them reduced \(\pmod p\). Then \(L=O(N^2\max(N,A)\max(N,B)p^{-\delta})\). Further they find an approximation of \(\sum_{m=1}^M(R_2-|I|)^2\) by applying related trigonometric sums.
scientific article; zbMATH DE number 1820763