Sumsets of sparse sets

Arithmetic Progressions in Sparse Sumsets

In this paper we show that sumsets A + B of finite sets A and B of integers, must contain long arithmetic progressions. The methods we use are completely elementary, in contrast to other works, which often rely on harmonic analysis. –Dedicated to Ron Graham on the occasion of his 70 birthday

ARITHMETIC PROGRESSIONS IN SPARSE SUMSETS Dedicated to Ron Graham on the occasion of his 70 birthday

In this paper we show that sumsets A + B of finite sets A and B of integers, must contain long arithmetic progressions. The methods we use are completely elementary, in contrast to other works, which often rely on harmonic analysis.

k-Term Arithmetic Progressions in Very Sparse Sumsets

One of the main focuses in combinatorial (and additive) number theory is that of “understanding” the structure of the sumset A + B = {a + b : a ∈ A, b ∈ B}, given certain information about the sets A and B. For example, one such problem is to determine the length of the longest arithmetic progression in this sumset, given that A,B ⊆ {0, 1, 2, ..., N} and |A|, |B| > δN , for some 0 < δ ≤ 1. The ...

On Sets Free of Sumsets with Summands of Prescribed Size

We study extremal problems about sets of integers that do not contain sumsets with summands of prescribed size. We analyse both finite sets and infinite sequences. We also study the connections of these problems with extremal problems of graphs and hypergraphs.

The additive structure of the squares inside rings

When defining the amount of additive structure on a set it is often convenient to consider certain sumsets; Calculating the cardinality of these sumsets can elucidate the set’s underlying structure. We begin by investigating finite sets of perfect squares and associated sumsets. We reveal how arithmetic progressions efficiently reduce the cardinality of sumsets and provide estimates for the min...

Sumsets of Sidon sets