Strong Uniform Distribution

Kit Nair (University of Liverpool)

Frank Adams Room 1, Alan Turing Building,

A sequences is strongly uniformly distributed if for bounded measurable \(f\) we have

\[ \lim_{N \to \infty} \frac{1}{N} \sum_{n=1}^{N} f(\{x_n\}) \to \int_0^1 f(x) dx\]

 The study of S.U.D. begins with a 1923 question of A. Khinchin.  He asked whether it is the case that if for a set \(B\)  contained in the unit interval of positive Lebesgue measure we always have

\[ \lim_{N \to \infty} \frac{1}{N} \sum_{n=1}^{N} \chi_B(\{nx\}) \to \lvert B \rvert \]

for almost all \(x\) with respect to Lebesgue measure. The answer is no as shown by J. M. Marstrand in a 1970 paper. In this talk I will discuss what questions are still current in this subject and what progress has been made. The talk will involve ideas from ergodic theory.

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