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Topics in descriptive set theory related to number theory and analysis


Ki, Haseo (1995) Topics in descriptive set theory related to number theory and analysis. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/SQV8-S991.


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 D2Π03 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 Σ03-complete and Π03-complete respectively. In particular, U provides a rare example of a natural Σ03-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 Σ03 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 Π11 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.

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):
  • Kechris, Alexander S.
Thesis Committee:
  • Unknown, Unknown
Defense Date:15 March 1995
Non-Caltech Author Email:haseo (AT)
Funding AgencyGrant Number
NSFDMS- 9020153
Record Number:CaltechETD:etd-10112007-111738
Persistent URL:
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:4040
Deposited By: Imported from ETD-db
Deposited On:12 Oct 2007
Last Modified:21 Dec 2019 02:51

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