Autor(es)

Verónica Becher and Yann Bugeaud and Theodore A Slaman

URL Pública

Abstract

Let be an integer greater than or equal to . A real number is simply normal to base if in its base- expansion every digit occurs with the same frequency . Let be the set of positive integers that are not perfect powers, hence is the set . Let be a function from to sets of positive integers such that, for each in , if is in then each divisor of is in and if is infinite then it is equal to the set of all positive integers. These conditions on are necessary for there to be a real number which is simply normal to exactly the bases such that is in and is in . We show these conditions are also sufficient and further establish that the set of real numbers that satisfy them has full Hausdorff dimension. This extends a result of W. M. Schmidt (1961/1962) on normal numbers to different bases.