Topics in descriptive set theory related to number theory and analysis

Mathematics;Set theory

Ki, Haseo ; Kechris, Alexander S.

University:California Institute of Technology

Department:Physics, Mathematics and Astronomy

关键词: Mathematics;    Set theory;   

Others  :  https://thesis.library.caltech.edu/4040/5/Ki_h_1995.pdf

美国|英语

来源: Caltech THESIS

【 摘 要 】

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.