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Expectation, and Its Connection with Quadratic Fields

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Probabilistic Diophantine Approximation

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Abstract

The diophantine sum

$$\displaystyle{ S_{\alpha }(n) =\sum _{ k=1}^{n}\left (\{k\alpha \} -\frac{1} {2}\right ) }$$
(2.1)

introduced in Sect. 1.2 [see (1.43)] is highly irregular as n → ∞, but its mean value

$$\displaystyle{ M_{\alpha }(N) = \frac{1} {N}\sum _{n=1}^{N}S_{\alpha }(n) }$$
(2.2)

exhibits a particularly simple and elegant asymptotic behavior for quadratic irrationals.

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Authors and Affiliations

  1. Department of Mathematics, Rutgers University, Piscataway, NJ, USA

    József Beck

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  1. József Beck
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Beck, J. (2014). Expectation, and Its Connection with Quadratic Fields. In: Probabilistic Diophantine Approximation. Springer Monographs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-10741-7_2

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