Topics in descriptive set theory related to number theory and analysis

• Main Author: Ki, Haseo
• Format: Others
• Published: 1995
• Online Access: https://thesis.library.caltech.edu/4040/5/Ki_h_1995.pdf; 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 <https://resolver.caltech.edu/CaltechETD:etd-10112007-111738>

<p>Based on the point of view of descriptive set theory, we have investigated several definable sets from number theory and analysis.</p> <p>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.</p> <p>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.</p> <p>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.</p> <p>In Chapter 4 we use techniques from number theory and the theory of Borel equivalence relations to provide a class of complete Π¹₁ sets.</p> <p>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.</p>