The Dilate Rollercoaster
Abstract
Given a finite subset A⊆ℤ and a multiset of non-zero integers Λ={λ1, …, λk}, the dilate sum Λ· A:=λ1· A+λ2· A+⋯+λk· A ={λ1a1+⋯+λkak: a1, …, ak∈ A}. For a non-zero real number r, say that Λ2 r-dominates Λ1 if we can partition some of the elements of Λ2 into blocks summing to the elements of r·Λ1. Then for all finite subsets A⊆ℤ, Λ2· A contains a translate of r·(Λ1· A), and hence |Λ1· A|≤|Λ2· A| for all A. In a recent paper, Kravitz (arXiv:2606.27087) proves the converse: if Λ2 does not r-dominate Λ1 for any r, then there exists a finite set A satisfying |Λ1· A|>|Λ2· 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⊆ℤ for which |A+7· A|>|2· A+3· A+5· A+7· A|. We extend Kravitz's method to apply it to multiple inequalities and multiple sets, showing that for multisets Λi,j with 1≤ i≤ k and 1≤ j≤ℓ, there exist finite sets A1, …, Aℓ⊆ℤ such that for all 1≤ i<k |Λi,1· A1+⋯+Λi,ℓ· Aℓ| >|Λi+1,1· A1+⋯+Λi+1,ℓ· Aℓ| provided there do not exist indices 1≤ i1<i2≤ k and a non-zero scalar r such that Λi2,j r-dominates Λi1,j for all 1≤ j≤ℓ. 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.
BibTeX
@misc{pebody2026dilate,
title = {{The Dilate Rollercoaster}},
author = {Luke Pebody},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.00217},
eprint = {2611.00217},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.00217},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Dilate Rollercoaster
AU - Luke Pebody
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.00217
UR - https://arxiv.org/abs/2611.00217
AB - Given a finite subset A⊆ℤ and a multiset of non-zero integers Λ={λ_1, …, λ_k}, the dilate sum Λ· A:=λ_1· A+λ_2· A+⋯+λ_k· A ={λ_1a_1+⋯+λ_ka_k: a_1, …, a_k∈ A}. For a non-zero real number r, say that Λ_2 r-dominates Λ_1 if we can partition some of the elements of Λ_2 into blocks summing to the elements of r·Λ_1. Then for all finite subsets A⊆ℤ, Λ_2· A contains a translate of r·(Λ_1· A), and hence |Λ_1· A|≤|Λ_2· A| for all A. In a recent paper, Kravitz (arXiv:2606.27087) proves the converse: if Λ_2 does not r-dominate Λ_1 for any r, then there exists a finite set A satisfying |Λ_1· A|>|Λ_2· 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⊆ℤ for which |A+7· A|>|2· A+3· A+5· A+7· A|. We extend Kravitz's method to apply it to multiple inequalities and multiple sets, showing that for multisets Λ_i,j with 1≤ i≤ k and 1≤ j≤ℓ, there exist finite sets A_1, …, A_ℓ⊆ℤ such that for all 1≤ i<k |Λ_i,1· A_1+⋯+Λ_i,ℓ· A_ℓ| >|Λ_i+1,1· A_1+⋯+Λ_i+1,ℓ· A_ℓ| provided there do not exist indices 1≤ i_1<i_2≤ k and a non-zero scalar r such that Λ_i_2,j r-dominates Λ_i_1,j for all 1≤ j≤ℓ. 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.
ER -