On distinct consecutive $r$-differences

Abstract

Suppose $A\subset \mathbb{R}$ of size $k$ has distinct consecutive $r$--differences, that is for $1 \leq i \leq k -r$, the $r$--tuples $$(a_{i+1} - a_i , \ldots , a_{i+r} - a_{i + r -1})$$ are distinct. Then for any finite $B \subset \mathbb{R}$, one has $$|A+B| \gg_r |A||B|^{1/(r+1)}.$$ Utilizing de Bruijn sequences, we show this inequality is sharp up to the constant. Moreover, for the sequence $\{n\alpha\}$, a sharp upper bound for the size of the distinct consecutive $r$--differences is obtained, which generalizes Steinhaus' three gap theorem. A dual problem on the consecutive $r$--differences of the returning times for some $\phi \in \mathbb{R}$ defined by $\{T : \{T\theta\}<\phi\}$ is also considered, which generalizes a result of Slater.