Citation
Ki, Haseo (1995) Topics in descriptive set theory related to number theory and analysis. Dissertation (Ph.D.), California Institute of Technology. http://resolver.caltech.edu/CaltechETD:etd10112007111738
Abstract
Based 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 D^{2}Π^{0}_{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 Usets from Mahler's classification of complex numbers. We are able to prove that U and T are Σ^{0}_{3}complete and Π^{0}_{3}complete respectively. In particular, U provides a rare example of a natural Σ^{0}_{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 Σ^{0}_{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 Π^{1}_{1} sets.
Finally, in Chapter 5, we solve a problem due to Ajtai and Kechris. For each differentiable function f on the unit circle, the KechrisWoodin 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 KechrisWoodin rank is finer than the Zalcwasser rank.
Item Type:  Thesis (Dissertation (Ph.D.))  

Subject Keywords:  Mathematics ; Set theory  
Degree Grantor:  California Institute of Technology  
Division:  Physics, Mathematics and Astronomy  
Major Option:  Mathematics  
Thesis Availability:  Public (worldwide access)  
Research Advisor(s): 
 
Thesis Committee: 
 
Defense Date:  15 March 1995  
NonCaltech Author Email:  haseo (AT) yonsei.ac.kr  
Funders: 
 
Record Number:  CaltechETD:etd10112007111738  
Persistent URL:  http://resolver.caltech.edu/CaltechETD:etd10112007111738  
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided.  
ID Code:  4040  
Collection:  CaltechTHESIS  
Deposited By:  Imported from ETDdb  
Deposited On:  12 Oct 2007  
Last Modified:  14 Apr 2015 18:44 
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