On Normal Numbers

BECHER, VERÓNICA

Helsinki

Congreso; 15th Congress of Logic, Methodology and Philosophy of Science; 2015

University of Helsinki

On normal numbers Flip a coin a large number of times and roughly half of the flips will come up heads and half will come up tails. Normality makes similar assertions about the digits in the expansions of a real number. For b an integer greater than or equal to 2, a real number x is simply normal to base b if every digit d in {0, 1, . . . , b-1} occurs in the base b expansion of x with asymptotic frequency 1/b (in the above example with coin tosses consider b equal to 2); a real number x is normal to base b if it is simply normal to all powers of b; and a real number x is absolutely normal if it is simply normal to all integer bases greater than or equal to 2. More than one hundred years ago E. Borel showed that almost all (for Lebesgue measure) real numbers are absolutely normal, and he asked for one example. He would have liked some fundamental mathematical constant such as pi or e, but this remains as the most famous open problem on normality. As for other examples, there have been several constructions of normal numbers since Borel's time, with varying levels of effectivity (computability). I will summarize the latest results, including our constructions of numbers normal to selected bases, a fast algorithm to compute an absolutely normal number which runs in nearly quadratic time, and an algorithm to compute an absolutely normal Liouville number.