ICC   25427
INSTITUTO DE INVESTIGACION EN CIENCIAS DE LA COMPUTACION
artículos
Título:
Normal numbers with digit dependencies
Autor/es:
VERĂ“NICA BECHER; CHRISTOPH AISTLEITNE; OLIVIER CARTON
Revista:
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY
Editorial:
AMER MATHEMATICAL SOC
Referencias:
Lugar: Providence; Año: 2019 vol. 113 p. 169 - 178
ISSN:
0002-9947
Resumen:
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 \$a_1a_2 ldots \$ of symbols from \${0, ldots, b-1}\$ such that \$a_n\$ is equal to \$a_{pn}\$, for every \$p\$ in \$P\$ and \$n=1,2,ldots\$. 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 \$T_P\$ is normal to all integer bases. We also consider the Toeplitz transform which maps the set of all sequences to the set \$T_P\$ and we characterize the normal sequences whose Toeplitz transform is normal as well.