Geodesic
On the construction of absolutely normal numbers
Christoph Aistleitner 1   ; Verónica Becher 2   ; Adrian-Maria Scheerer 1   ; Theodore A. Slaman 3
1 Institute of Analysis and Number Theory Graz University of Technology A-8010 Graz, Austria
2 Departamento de Computación Facultad de Ciencias Exactas y Naturales Universidad de Buenos Aires & ICC, CONICET Pabellón I, Ciudad Universitaria C1428EGA Buenos Aires, Argentina
3 Department of Mathematics University of California Berkeley 719 Evans Hall #3840 Berkeley, CA 94720-3840, U.S.A.
Acta Arithmetica, Tome 180 (2017) no. 4, pp. 333-346 Cet article a éte moissonné depuis la source Institute of Mathematics Polish Academy of Sciences

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We give a construction of an absolutely normal real number $x$ such that for every integer $b \ge 2$, the discrepancy of the first $N$ terms of the sequence $(b^n x \mod 1)_{n\geq 0}$ is of asymptotic order $\mathcal{O}(N^{-1/2})$. 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.
DOI : 10.4064/aa170213-5-8
Keywords: construction absolutely normal real number nbsp every integer discrepancy first terms sequence mod geq asymptotic order mathcal below order discrepancy which holds almost real numbers even existence absolutely normal numbers having discrepancy small asymptotic order has known before

Christoph Aistleitner  1   ; Verónica Becher  2   ; Adrian-Maria Scheerer  1   ; Theodore A. Slaman  3

1 Institute of Analysis and Number Theory Graz University of Technology A-8010 Graz, Austria
2 Departamento de Computación Facultad de Ciencias Exactas y Naturales Universidad de Buenos Aires & ICC, CONICET Pabellón I, Ciudad Universitaria C1428EGA Buenos Aires, Argentina
3 Department of Mathematics University of California Berkeley 719 Evans Hall #3840 Berkeley, CA 94720-3840, U.S.A. 
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     title = {On the construction of absolutely normal numbers},
     journal = {Acta Arithmetica},
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Christoph Aistleitner; Verónica Becher; Adrian-Maria Scheerer; Theodore A. Slaman. On the construction of absolutely normal numbers. Acta Arithmetica, Tome 180 (2017) no. 4, pp. 333-346. doi: 10.4064/aa170213-5-8

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