On the construction of absolutely normal numbers

Christoph Aistleitner; Adrian-Maria Scheerer; Veronica Becher; Theodore Slaman

doi:10.4064/aa170213-5-8

On the construction of absolutely normal numbers

Christoph Aistleitner, Adrian-Maria Scheerer, Veronica Becher, Theodore Slaman

Institute of Analysis and Number Theory (5010)

Research output: Contribution to journal › Article › peer-review

Abstract

We give a construction of an absolutely normal real number x such that for every integer b≥2, the discrepancy of the first N terms of the sequence (bnxmod1)n≥0 is of asymptotic order O(N−1/2). This is below the order of discrepancy which holds for almost all real numbers. Even the existence of absolutely normal numbers having a discrepancy of such a small asymptotic order has not been known before.

Original language	English
Pages (from-to)	333-346
Journal	Acta Arithmetica
Volume	180
Issue number	4
DOIs	https://doi.org/10.4064/aa170213-5-8
Publication status	Published - 2017

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10.4064/aa170213-5-8

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title = "On the construction of absolutely normal numbers",

abstract = "We give a construction of an absolutely normal real number x such that for every integer b≥2, the discrepancy of the first N terms of the sequence (bnxmod1)n≥0 is of asymptotic order O(N−1/2). This is below the order of discrepancy which holds for almost all real numbers. Even the existence of absolutely normal numbers having a discrepancy of such a small asymptotic order has not been known before.",

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AU - Becher, Veronica

AU - Slaman, Theodore

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AB - We give a construction of an absolutely normal real number x such that for every integer b≥2, the discrepancy of the first N terms of the sequence (bnxmod1)n≥0 is of asymptotic order O(N−1/2). This is below the order of discrepancy which holds for almost all real numbers. Even the existence of absolutely normal numbers having a discrepancy of such a small asymptotic order has not been known before.

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DO - 10.4064/aa170213-5-8

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SP - 333

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JO - Acta Arithmetica

JF - Acta Arithmetica

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On the construction of absolutely normal numbers

Abstract

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