Nombres normaux
Nous rassemblons divers résultats sur les nombres normaux et en déduisons de nouveaux résultats.
Nombres normaux dans diverses bases
En s’inspirant d’un article de Feldman et Smorodinsky on étudie l’apparition d’un bloc de chiffres fixé dans le -développement de . On montre que si et sont des nombres de Pisot non équivalents, les ensembles des nombres normaux au sens des chiffres pour et sont différents, et que si est un Pisot et un entier algébrique non équivalent à , les ensembles des nombres géométriquement normaux relativement à et sont distincts.
Nombres normaux et processus déterministes
Non literal tranducers and some problems of normality
A new proof of Maxfield’s theorem is given, using automata and results from Symbolic Dynamics. These techniques permit to prove that points that are near normality to base (resp. ) are also near normality to base (resp. ), and to study genericity preservation for non Lebesgue measures when going from one base to the other. Finally, similar results are proved to bases the golden mean and its square.
Non-primes are recursively divisible.
Normal number constructions for Cantor series with slowly growing bases
Let be a sequence of bases with . In the case when the are slowly growing and satisfy some additional weak conditions, we provide a construction of a number whose -Cantor series expansion is both -normal and -distribution normal. Moreover, this construction will result in a computable number provided we have some additional conditions on the computability of , and from this construction we can provide computable constructions of numbers with atypical normality properties.
Normal numbers and the Borel hierarchy
We show that the set of absolutely normal numbers is Π⁰₃-complete in the Borel hierarchy of subsets of real numbers. Similarly, the set of absolutely normal numbers is Π⁰₃-complete in the effective Borel hierarchy.
Normal numbers and the middle prime factor of an integer
Let pₘ(n) stand for the middle prime factor of the integer n ≥ 2. We first establish that the size of log pₘ(n) is close to √(log n) for almost all n. We then show how one can use the successive values of pₘ(n) to generate a normal number in any given base D ≥ 2. Finally, we study the behavior of exponential sums involving the middle prime factor function.
Normal recurring decimals, normal periodic systems, (j,ε) - normality, and normal numbers
Normalität bezüglich Matrizen.
Normality of numbers generated by the values of polynomials at primes
Number theoretic expansions, algorithms and metrical observations.