A Journey Through Normality

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They say the only normal people are the ones you don't know very well. But what about numbers? Which ones are normal, and how well do we really know them? A real number x is normal in base b if every block of digits of the same length appears in the b-adic digit expansion of x with equal limiting frequency. While the concept is easy to define, proving that specific numbers are normal often requires deep and intricate methods. The study of normal numbers connects diverse areas of mathematics, including harmonic analysis, ergodic theory, Diophantine approximation, and fractal geometry.

In this talk, I will survey key results and techniques in the study of normal numbers, explore the challenges posed by long-standing open problems, and present the recent resolution of a conjecture by Kahane and Salem regarding the properties of partially normal numbers. This work draws on methods from Fourier analysis, geometric measure theory, and probabilistic number theory. Joint work with Junqiang Zhang.

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