Topics in descriptive set theory related to number theory and analysis

Citation

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. https://resolver.caltech.edu/CaltechETD:etd-10112007-111738

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 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.

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