Lecture in Harmonic Analysis
The rise and fall of Cantor and the Salem-Zygmund theorem
Cantor - the early years
Consider a series of trigonometric functions that converges everywhere. Are the coefficients uniquely determined by the limit function? A young Cantor was tasked with solving this difficult question, which he successfully answered in the affirmative in 1870. What happens, he then wondered, when the series converges "less than everywhere"? Cantor improved his previous result, but in doing so, contradicted a widely believed notion - he paradoxically concluded that the interval [0, 1] cannot be countable! This was the motivation for his invention of modern set theory.
The Salem-Zygmund theorem
The problem of classifying how much "less than everywhere" the convergence may be while still ensuring uniqueness is open still today. In 1916, surprisingly, almost everywhere convergence was shown to not ensure uniqueness. In 1955, another breakthrough was made: A generalized construction of the Cantor set (with ratio other than the classical 1/3) was shown by Salem and Zygmund to either ensure or not ensure uniqueness, depending on arithmetic properties of the ratio used to build the set. This revelation sheds some light on the complexity of the classification problem.
In my lecture, we will quickly follow the origin of Cantor's downfall to his eventual insanity, and then focus on analyzing sets of uniqueness and hopefully prove (at least partially) the Salem-Zygmund theorem.
Required knowledge: This is a graduate-level lecture, but with regard to the field of advanced harmonic analysis/descriptive set theory, it will be introductory and newbie-friendly. You should know a little Fourier analysis, a little Galois theory, a little topology, and have a little familiarity with the (simple) Cantor set.