Multivariate normal distribution for integral points on varieties

El-Baz, Daniel; Loughran, Daniel; Sofos, Efthymios

doi:10.1090/tran/8545

Multivariate normal distribution for integral points on varieties
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by Daniel El-Baz, Daniel Loughran and Efthymios Sofos PDF

Trans. Amer. Math. Soc. 375 (2022), 3089-3128 Request permission

Abstract:

Given a variety with coefficients in $\mathbb {Z}$, we study the distribution of the number of primes dividing the coordinates as we vary an integral point. Under suitable assumptions, we show that this has a multivariate normal distribution. We generalise this to more general Weil divisors, where we obtain a geometric interpretation of the covariance matrix. For our results we develop a version of the Erdős–Kac theorem that applies to fairly general integer sequences and does not require a positive exponent of level of distribution.

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