On distinct consecutive $r$-differences

Suppose $A \subset R$ 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 ${n α}$ , 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 $ϕ \in R$ defined by ${T : {T θ} < ϕ}$ is also considered, which generalizes a result of Slater.

Suppose $A \subset R$ 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 ${n α}$ , 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 $ϕ \in R$ defined by ${T : {T θ} < ϕ}$ is also considered, which generalizes a result of Slater.

On distinct consecutive $r$ -differences