Iterated sums races

Abstract: This is joint work with Paul P\' eringuey. \\ Our work provides a solution to a question posed by M. Nathanson in late 2024, but we later realized that this problem, along with an even more challenging one, had already been solved by N. Kravitz in a paper posted on arXiv in January 2025. While our construction is similar to his, it is simpler, and we hope that it can serve as an introductory step toward understanding the underlying ideas. \\ Nathanson's question is as follows: \\ \textit{For every integer $m \geq 3$, do there exist finite sets $A$ and $B$ of integers and an increasing sequence of positive integers $h_1 < h_2 < \cdots < h_m$, such that: \\ $$ |h_i A| > |h_i B| \quad \text{if } i \text{ is odd,} $$ $$ |h_i A| < |h_i B| \quad \text{if } i \text{ is even.} $$ \\ Additionally, do there exist such sets with $|A| = |B|$? Can such sets be constructed with $|A| = |B|$ and $\text{diam}(A) = \text{diam}(B)$?} \\ We provide a positive answer to these questions and propose an iterative construction of sets that satisfy these conditions.

Mathematics

Audience: researchers in the topic

Combinatorial and additive number theory (CANT 2025)