Topics in descriptive set theory related to number theory and analysis
<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 D<sup&g...
| Summary: | <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 D<sup>2</sup>Π<sup>0</sup><sub>3</sub> 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 Σ<sup>0</sup><sub>3</sub>-complete and Π<sup>0</sup><sub>3</sub>-complete respectively. In particular, U provides a rare example of a natural Σ<sup>0</sup><sub>3</sub>-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 Σ<sup>0</sup><sub>3</sub> 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 Π<sup>1</sup><sub>1</sub> 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>
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