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:etd-10112007-111738
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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 [...] 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 [...]-complete and [...]-complete respectively. In particular, U provides a rare example of a natural [...]-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 [...] 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 [...] 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.))|
|Degree Grantor:||California Institute of Technology|
|Division:||Physics, Mathematics and Astronomy|
|Thesis Availability:||Restricted to Caltech community only|
|Defense Date:||15 March 1995|
|Default Usage Policy:||No commercial reproduction, distribution, display or performance rights in this work are provided.|
|Deposited By:||Imported from ETD-db|
|Deposited On:||12 Oct 2007|
|Last Modified:||26 Dec 2012 03:04|
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