Departmental Colloquium

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 a given base b if every block of digits of the same length appears with equal limiting frequency in the b-adic digit expansion of x. While the definition is simple, proving that specific numbers are normal often requires deep and intricate methods. The study of normality weaves together ideas from 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, highlight major open problems, and discuss recent progress. In particular, I will present the resolution of a conjecture by Kahane and Salem concerning the normality of certain random series, a result that draws on tools from Fourier analysis and probabilistic number theory. This is based on joint work with Junqiang Zhang.