Uniform Distribution and Rigidity

A sequence in [0,1] is uniformly distributed if for all continuous functions F, the averages of F at the points in the sequence converge to the integral of F on [0,1]. We know how to construct lots of such sequences. A suitably randomly chosen sequence will work. But many very simple sequences work too. For example, for any irrational number r, the fractional parts of the arithmetic sequence (kr : 1,2,3,…) are uniformly distributed. It is not so easy to see, but we also know there are MANY r > 1 such that the fractional parts of the kth powers of r are uniformly distributed. But we do not know of any concrete example of r with this property! It has long been conjectured hat r = 3/2 will work. Can we figure out if this is true or false? On the other hand, there are values of r such that the fractional parts of the kth powers of r are not uniformly distributed. For example, r = 1/2 + 5^{1/2}/2. Why is this? Can we characterize this behavior? In contrast, very different behavior goes on if we do not use all the whole numbers in our sequences. Suppose we fix an increasing sequence (n(k) : k=1,2,3,…). Can we describe, or even just find, values of r such that the fractional parts of the sequence (n(k) r : k=1,2,3,…) are also uniformly distributed? In direct contrast to uniform distribution, we do not understand very well the rigidity of sequences like the ones above. That is, fix an increasing sequence (n(k) : k=1,2,3,…). What are the values of r such that the fractional parts of n(k)r go to 0 as k goes to infinity? Are there any? How does it depend on how fast n(k) is increasing? For example, are there any irrational numbers r such that the fractional parts of (exp(kr) : k=1,2,3,…) go to 0 as k goes to infinity? What about something a little faster like replacing k by a polynomial function of k? As part of this project, we will explore many interesting facts and conjectures related to uniform distribution and rigidity. These involve an interplay of number theory, harmonic analysis, and dynamical systems. We will seek to understand what we know, and better understand what we do not know, about this interplay. We will not only use existing techniques and results, but we will also see what we can do with numerical experiments to explore the behavior of sequences modulo one, especially where we do not have theorems that adequately explain the behavior of the sequences.

For more details see here.