On distinct consecutive $r$-differences

Abstract

Suppose $A\subset \mathbb{R}$ of size $k$ has distinct consecutive $r$--differences, that is for $1\le i\le k-r$, the $r$--tuples $$(a_{i+1}-a_i,\dots ,a_{i+r}-a_{i+r-1})$$ 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 $\varphi \in \mathbb{R}$ defined by $\{T:\{T\theta \}<\varphi \}$ is also considered, which generalizes a result of Slater.

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