On distinct consecutive differences

TL;DR

This paper proves a lower bound on the size of the sumset A+B when A has distinct consecutive differences, showing that the sumset grows at least proportionally to the square root of |A| times |B|, and that this bound is tight.

Contribution

It establishes a tight lower bound on sumset size for sets with distinct consecutive differences, extending additive combinatorics knowledge.

Findings

|A+B|  |A|^{1/2}|B| for sets with distinct consecutive differences

The bound is tight up to a constant factor

Provides insight into sumset growth in additive combinatorics

Abstract

We show that if $A=\{a_1 < a_2 < \ldots < a_k\}$ is a set of real numbers such that the differences of the consecutive elements are distinct, then for and finite $B \subset \mathbb{R}$, $$|A+B|\gg |A|^{1/2}|B|.$$ The bound is tight up to the constant.