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What Is “Probabilistic” Diophantine Approximation?

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Book cover Probabilistic Diophantine Approximation

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Abstract

We discuss some surprising new developments concerning \(\sqrt{ 2}\), and in general the class of quadratic irrationals. We use \(\sqrt{ 2}\) as the representative for the whole class. These results provide some rigorous evidence for a mysterious general phenomenon that we call the Giant Leap. In a nutshell, it is about the unexpected randomness of explicit sequences (Giant Leap to full-blown randomness). The reader may jump ahead to Sect. 2.5 for a detailed discussion of this issue.

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Beck, J. (2014). What Is “Probabilistic” Diophantine Approximation?. In: Probabilistic Diophantine Approximation. Springer Monographs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-10741-7_1

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