Solutions to several problems on infinite sumset configurations
Abstract
We resolve a number of the problems collected by Kra, Moreira, Richter and Robertson in their survey "Problems on infinite sumset configurations in the integers and beyond". Our main result is that in every countably infinite abelian group, every set of positive upper Banach density contains a sumset B1+…+Bk of k infinite sets, for every k; the proof uses the Kronecker factor, but no higher-order Host-Kra structure theory, nilsystems of step greater than one, or nilspaces. We also answer, positively or negatively, questions on monochromatic sets B+mB, on sumsets in sets of logarithmic density, on polynomial sumset configurations, on sums of the form Bi1+…+Bim with Bi drawn from a density family, on random sets, on ordered sumsets in shifted primes, on sums and products in ℕ and ℚ, and on sumsets in abelian groups. Several of these answers concern the questions as literally stated, and two follow directly from known examples; in each case we say so and discuss the natural corrected question. Finally, answering a question of Kousek and Radić, we show that the supremum of the lower Banach densities of sets A⊂ℕ such that B+B+tnot⊂ A for every infinite B⊂ℕ and every t≥0 (respectively B+Bnot⊂ A for every infinite B) is 1/2 (respectively 3/4). The main theorem and most of the other results are formalized in Lean 4, with some published theorems entering as explicit hypotheses.
Related
- Infinite Sum-Product Configurations in Parallel
- On several judicious partitioning problems of graphs and hypergraphs
- A new class of pyramidal central configurations for spatial Newtonian 6-body problems
- Deformation of sine-Gordon two-soliton solutions in φ4 kink-antikink configurations
- Polynomial mixing for infinite-energy solutions of the white-forced Navier--Stokes equation in an infinite pipe
BibTeX
@misc{li2026solutions,
title = {{Solutions to several problems on infinite sumset configurations}},
author = {Shisheng Li and Yong Su},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.39637},
eprint = {2610.39637},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.39637},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Solutions to several problems on infinite sumset configurations
AU - Shisheng Li
AU - Yong Su
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.39637
UR - https://arxiv.org/abs/2610.39637
AB - We resolve a number of the problems collected by Kra, Moreira, Richter and Robertson in their survey "Problems on infinite sumset configurations in the integers and beyond". Our main result is that in every countably infinite abelian group, every set of positive upper Banach density contains a sumset B_1+…+B_k of k infinite sets, for every k; the proof uses the Kronecker factor, but no higher-order Host-Kra structure theory, nilsystems of step greater than one, or nilspaces. We also answer, positively or negatively, questions on monochromatic sets B+mB, on sumsets in sets of logarithmic density, on polynomial sumset configurations, on sums of the form B_i_1+…+B_i_m with B_i drawn from a density family, on random sets, on ordered sumsets in shifted primes, on sums and products in ℕ and ℚ, and on sumsets in abelian groups. Several of these answers concern the questions as literally stated, and two follow directly from known examples; in each case we say so and discuss the natural corrected question. Finally, answering a question of Kousek and Radić, we show that the supremum of the lower Banach densities of sets A⊂ℕ such that B+B+tnot⊂ A for every infinite B⊂ℕ and every t≥0 (respectively B+Bnot⊂ A for every infinite B) is 1/2 (respectively 3/4). The main theorem and most of the other results are formalized in Lean 4, with some published theorems entering as explicit hypotheses.
ER -