Low discrepancy sequences failing Poissonian pair correlations

03/05/2019

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by   Verónica Becher, et al.

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M. Levin defined a real number x that satisfies that the sequence of the fractional parts of (2^n x)_n≥ 1 are such that the first N terms have discrepancy O(( N)^2/ N), which is the smallest discrepancy known for this kind of parametric sequences. In this work we show that the fractional parts of the sequence (2^n x)_n≥ 1 fail to have Poissonian pair correlations. Moreover, we show that all the real numbers x that are variants of Levin's number using Pascal triangle matrices are such that the fractional parts of the sequence (2^n x)_n≥ 1 fail to have Poissonian pair correlations.