ICC   25427
INSTITUTO DE INVESTIGACION EN CIENCIAS DE LA COMPUTACION
Unidad Ejecutora - UE
congresos y reuniones científicas
Título:
Normal Numbers with Digit Dependencies
Autor/es:
CARTÓN, OLIVIER; VERÓNICA BECHER; CHRISTOPH AISTLEITNER
Lugar:
Singapur
Reunión:
Workshop; Equidistribution: Arithmetic, Computational and Probabilistic As- pects. Institute for Mathematical Sciences, National University Singapore, 29 April-May 17, 2019.; 2019
Institución organizadora:
Institute for Mathematical Sciences, National University Singapore,
Resumen:
Abstract. We give metric theorems for the property of Borel normality for real numbers under the assumption of digit dependencies in their expansion in a given integer base. We quantify precisely how much digit dependence can be allowed such that, still, almost all real numbers are normal. Our theorem states that almost all real numbers are normal when at least slightly more than log log n consecutive digits with indices starting at position n are independent. As the main application, we consider the Toeplitz set T_P , which is the set of all sequences a1 a2... of symbols from {0,...,b−1} such that an is equal to a_{p_n} ,for every p in P and n=1,2,.... Here b is an integer base and P is a finite set of prime numbers. We show that almost every real number whose base b expansion is in T_P is normal to base b. In the case when P is the singleton set {2} we prove that more is true: almost every real number whose base b expansion is in TP is normal to all integer bases. We also consider the Toeplitz transform which maps the set of all sequences to the set TP and we characterize the normal sequences whose Toeplitz transform is normal as well.