# Distinct consecutive r-differences

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 $A \subset \mathbb{R}$ with special structure $|A+A| \gg |A|^{1 + \delta}$ for some positive $\delta$ (here we adopt Vinogradov’s notation). Note that $|A+A| = 2|A| -1$ for arithmetic progressions, so we really need to make assumptions about the structure of $A$.

Motivated by a paper of Solymosi, we introduce the notion of distinct consecutive $r$-differences. We say $A \subset \mathbb{R}$ has distinct consecutive $r$-differences if $(a_{i+1} - a_i , \ldots , a_{i+r} - a_{i +r -1})$ are distinct for $1 \leq i \leq |A|-r$.

Assume $A$ has distinct consecutive $r$-differences. We show that for any finite $B \subset \mathbb{R}$, one has $|A+B| \gg |A||B|^{1/(r+1)}$ and that this inequality is sharp in such generality. We wonder if one can improve upon this if $B= A$ and ask what is the largest $\theta_r$ such that $|A+A| \gg |A|^{1 + \theta_r/(r+1)}$. The above result implies $\theta_r \geq 1$, while in our paper we use de Bruijn sequences to show $\theta_r \leq 2$.

When $A$ has additive structure, the results from our paper suggest that $A$ should have few distinct consecutive $r$-differences. We investigate two of these cases show that these $A$ 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.