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Pair correlation of lattice points with prime constraint. (English) Zbl 1327.11053
Let \(\Omega\) be a be a star-shaped region in the plane bounded by a curve \(\mathcal C\), parametrized, for any \((x,y)\in\Omega\), by \(x = \rho_\Omega(\alpha)\cos(\alpha), y=\rho(\alpha)\sin(\alpha)\), where \(\rho_\Omega\) is continuous and piecewise \(C^1\) on \([0,2\pi]\). For each large integer \(X\), the dilated region \[ \Omega_X:=\{(x,y)\in\mathbb R^2: (x/X,y/Y)\in\Omega\} \] has an area equal to \(X^2\cdot Area(\Omega)\) and is bounded by the curve \[ \mathcal C_X = \{(x, y) \in \mathbb R^2 :(x/X,y/Y)\in\mathcal C). \] Given a pair of coprime integers \(a\) and \(b\), the authors define the set \[ \Omega_X^{(a,b)}=\{(x,y) \in\mathbb Z^2\cap\Omega_X:ax+by\text{ is a prime}\} \] where the prime may be positive or negative.
let \(0 <\theta_1 < \dots < \theta_N < 2\pi\) where \(N=N(\Omega,X)\) be the angles of straight lines joining the origin to points of \(\Omega_X^{(a,b)}\), taken without multiplicities, that correspond to elements from \(\Omega_X^{(a,b)}\). They are normalized to \( 0 <\theta_1/{2\pi}< \theta_2/{2\pi}<\dots <\theta_N/2\pi < 1.\) The pair correlation function measures is defined as the density of the differences between pairs of elements of a given sequence. For the sequence \((\theta_n/2\pi)_{n\leq N}\) in \([0, 1]\), the limiting pair correlation function \(R_2(\Omega_{a,b)},\lambda)\) is given, if it exists, by \[ \lim_{N\to\infty} \frac{1}{N}\#\Big\{ 1\leq n_1\not=n_2\leq N:\frac{\theta_{n_1}}{2\pi}-\frac{\theta_{n2}}{2\pi}\in \frac{1}{N}\{\mathbf I\}\Big\}=\int_{\mathbf I}R_2(\Omega_{(a,b)},\lambda)\,d\lambda \] for any interval \({\mathbf I}\subset \mathbb R\). The authors prove:
Theorem. As \(X\) goes to \(\infty\), the limiting pair correlation function \(R_2(\Omega_{(a,b)},\lambda)\) of the angles of straight lines from the origin to the set \(\Omega_X^{(a,b)}\) exists, is independent of \(a,b\) and is identically equal to the constant \(\frac{\pi}{2A(\Omega)^2}\int_0^{2\pi}\rho_\Omega(\theta)^4\,d\theta\) where \(A(\Omega)\) is the area of the region \(\Omega\).
The authors show that the pair correlation density they obtained for general \(\Omega\) corresponds to a Poisson process with non-uniform density.
The authors give an historic context in particular, points visible from the origine [F. P. Boca et al., Commun. Math. Phys. 213, No. 2, 433–470 (2000; Zbl 0989.11049)], correlations of directions of straight lines of lattice points from the origin [F. P. Boca and the second author, Trans. Am. Math. Soc. 358, No. 4, 1797–1825 (2006; Zbl 1154.11022)], correlations of zeros of zeta functions of principal \(L\)-functions and random matrix theory [N. M. Katz and P. Sarnak, Bull. Am. Math. Soc., New Ser. 36, No. 1, 1–26 (1999; Zbl 0921.11047); H. L. Montgomery, Proc. Symp. Pure Math. 24, St. Louis Univ. Missouri 1972, 181–193 (1973; Zbl 0268.10023); Z. Rudnick and P. Sarnak, Duke Math. J. 81, No. 2, 269–322 (1996; Zbl 0866.11050)].
11K06 General theory of distribution modulo \(1\)
11J71 Distribution modulo one
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