  We investigate additive properties of sets $A,$ where $A=\{a_1,a_2,\ldots
,a_k\}$ is a monotone increasing set of real numbers, and the differences of
consecutive elements are all distinct. It is known that $|A+B|\geq
c|A||B|^{1/2}$ for any finite set of numbers $B.$ The bound is tight up to the
constant multiplier. We give a new proof to this result using bounds on
crossing numbers of geometric graphs. We construct examples showing the limits
of possible improvements. In particular, we show that there are arbitrarily
large sets with different consecutive differences and sub-quadratic sumset
sizes.
