1 results
Sumsets of semiconvex sets
- Part of
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- Journal:
- Canadian Mathematical Bulletin / Volume 65 / Issue 1 / March 2022
- Published online by Cambridge University Press:
- 26 February 2021, pp. 84-94
- Print publication:
- March 2022
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- Article
- Export citation
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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 size.
