CaltechTHESISCitation Ki, Haseo (1995) Topics in descriptive set theory related to number theory and analysis. /CaltechETD:etd-10112007-111738AbstractBased on the point of view of descriptive set theory, we have investigated several definable sets from number theory and analysis. In Chapter 1 we solve two problems due to Kechris about sets arising in number theory, provide an example of a somewhat natural D2 3 set, and exhibit an exact relationship between the Borel class of a nonempty subset X of the unit interval and the class of subsets of N whose densities lie in X.
In Chapter 2 we study the A, S, T and U-sets from Mahler's classification of complex numbers. We are able to prove that U and T are 3-complete and 3-complete set.
In Chapter 3 we solve a question due to Kechris about UCF, the set of all continuous functions, on the unit circle, with Fourier series uniformly convergent. We further show that any 3 set, which contains UCF, must contain a continuous function with Fourier series divergent.
In Chapter 4 we use techniques from number theory and the theory of Borel equivalence relations to provide a class of complete 11 sets. Finally, in Chapter 5, we solve a problem due to Ajtai and Kechris.
For each differentiable function f on the unit circle, the Kechris-Woodin rank measures the failure of continuity of the derivative function f', while the Zalcwasser rank measures how close the Fourier series of f is to being a uniformly convergent series. We show that the Kechris-Woodin rank is finer than the Zalcwasser rank.