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.

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