Junxian Li, a fellow graduate student at UIUC, and I just uploaded our paper On distinct consecutive r-differences to the arXiv.
Our goal is to show for finite with special structure for some positive (here we adopt Vinogradov’s notation). Note that for arithmetic progressions, so we really need to make assumptions about the structure of .
Motivated by a paper of Solymosi, we introduce the notion of distinct consecutive -differences. We say has distinct consecutive -differences if are distinct for .
Assume has distinct consecutive -differences. We show that for any finite , one has and that this inequality is sharp in such generality. We wonder if one can improve upon this if and ask what is the largest such that . The above result implies , while in our paper we use de Bruijn sequences to show .
When has additive structure, the results from our paper suggest that should have few distinct consecutive -differences. We investigate two of these cases show that these have very few distinct consecutive differences. In the process, we generalize Steinhaus’ 3-gap theorem as well as a result of Slater concerning return times of irrational numbers under the multiplication map of the Torus.