Normality together with other properties

BECHER, VERÓNICA

Nancy

Seminario; Number Theory Seminar Nancy-Metz; 2017

Institut Élie Cartan de Lorraine,Université de Lorraine

Normality together with other propertiesVerónica Becher, Universidad de Buenos AiresAbstract: Let b be an integer greater than or equal to 2. A real number is called simply normal to base b if each digit 0,..,b-1 occurs in its b-ary expansion with the same frequency 1/b. It is called normal to base b if it is simply normal to every base b^k, for k=1,2,?, and it is called absolutely normal if it is normal to very integer base b. Borel proved that almost all real numbers, in the sense of the Lebesgue measure, are absolutely normal and he conjectured that the irrational algebraic numbers are absolutely normal. So far, no algebraic irrational has been proved nor disproved to be normal to any base. All known examples of absolutely normal numbers were obtained by constructions specifically made to comply the definition. These constructions were extended to obtain normality together with some other property having Lebesgue measure 1, such as continued fraction normality, normality to Pisot bases, or irrationality exponent equal to 2. For a few properties having Lebesgue measure 0, a construction is known that gives a normal number with this property, such as the construction of a number normal to all bases except to powers of 3, or the construction of an absolutely normal Liouville number.