Descriptive Complexity in Number Theory and Dynamics

Informally, a real number is normal in base b if in its b-ary expansion, all digits and blocks of digits occur as often as one would expect them to, uniformly at random. We will denote the set of numbers normal in base b by N(b). Kechris asked several questions involving descriptive complexity of sets of normal numbers. The first of these was resolved in 1994 when Ki and Linton proved that N(b) is Pi_3^0-complete. Further questions were resolved by Becher, Heiber, and Slaman who showed that Intersection_{b=2}^infty N(b) is Pi_3^0-complete and that Union_{b=2}^infty N(b) is Sigma_4^0-complete. Many of the techniques used in these proofs can be used elsewhere. We will discuss recent results where similar techniques were applied to solve a problem of Sharkovsky and Sivak and a question of Kolyada, Misiurewicz, and Snoha. Furthermore, we will discuss a recent result where the set of numbers that are continued fraction normal, but not normal in any base b, was shown to be complete at the expected level of D_2(Pi_3^0). An immediate corollary is that this set is uncountable, a result (due to Vandehey) only known previously assuming the generalized Riemann hypothesis.