Sumsets of Semiconvex sets

DOI10.4153/S0008439521000096arXiv2008.08021MaRDI QIDQ6347315

Publication date: 18 August 2020

Abstract: We investigate additive properties of sets

A,

where

A = a_{1}, a_{2}, l d o t s, 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 | g e q 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.

Mathematics Subject Classification ID

Combinatorial aspects of difference sets (number-theoretic, group-theoretic, etc.) (05B10) Density, gaps, topology (11B05) Additive bases, including sumsets (11B13) Arithmetic combinatorics; higher degree uniformity (11B30)

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