论文标题
Semiconvex集的总和
Sumsets of Semiconvex sets
论文作者
论文摘要
我们研究集合$ a的加性属性，其中$ a = \ {a_1，a_2，\ ldots，a_k \} $是单调增加的实数集，并且连续元素的差异都是不同的。众所周知，对于任何有限的数字$B。$ | a+b | \ geq c | a || b |^{1/2} $。我们使用几何图的交叉数量的界限为此结果提供了新的证明。我们构建了显示可能改进的局限性的示例。特别是，我们表明有不同的连续差异和次级集合大小的任意集。
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.
