arXiv:2611.00217v1 Announce Type: new Abstract: Given a finite subset $A\subseteq\mathbb{Z}$ and a multiset of non-zero integers $\Lambda=\{\lambda_1, \ldots, \lambda_k\}$, the dilate sum \[\Lambda\cdot A:=\lambda_1\cdot A+\lambda_2\cdot A+\cdots+\lambda_k\cdot A =\{\lambda_1a_1+\cdots+\lambda_ka_k: a_1, \ldots, a_k\in A\}.\] For a non-zero real number $r$, say that $\Lambda_2$ $r$-dominates $\Lambda_1$ if we can partition some of the elements of $\Lambda_2$ into blocks summing to the elements of $r\cdot\Lambda_1$. Then for all finite subsets $A\subseteq\mathbb{Z}$, $\Lambda_2\cdot A$ contains a translate of $r\cdot(\Lambda_1\cdot A)$, and hence $|\Lambda_1\cdot A|\le|\Lambda_2\cdot A|$ for all $A$. In a recent paper, Kravitz (arXiv:2606.27087) proves the converse: if $\Lambda_2$ does not $r$-dominate $\Lambda_1$ for any $r$, then there exists a finite set $A$ satisfying $|\Lambda_1\cdot A|>|\Lambda_2\cdot A|$. For example, $\{2, 3, 5, 7\}$ does not contain two disjoint subsets with a sum ratio of 7, and so there exists a finite $A\subseteq\mathbb{Z}$ for which $|A+7\cdot A|>|2\cdot A+3\cdot A+5\cdot A+7\cdot A|$. We extend Kravitz's method to apply it to multiple inequalities and multiple sets, showing that for multisets $\Lambda_{i,j}$ with $1\le i\le k$ and $1\le j\le\ell$, there exist finite sets $A_1, \ldots, A_\ell\subseteq\mathbb{Z}$ such that for all $1\le i<k$ \[|\Lambda_{i,1}\cdot A_1+\cdots+\Lambda_{i,\ell}\cdot A_\ell| >|\Lambda_{i+1,1}\cdot A_1+\cdots+\Lambda_{i+1,\ell}\cdot A_\ell|\] provided there do not exist indices $1\le i_1<i_2\le k$ and a non-zero scalar $r$ such that $\Lambda_{i_2,j}$ $r$-dominates $\Lambda_{i_1,j}$ for all $1\le j\le\ell$. Along the way, we resolve the analogue of Nathanson's Problem 1 for his Theorem 4 (arXiv:1604.08940), as well as the problems in Section 1.4.3 of Kravitz's paper.
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