Adherence Semigroups and Density Finite-Sums Configurations

2026/07/25 by Song Shao, Hui Xu
#math.DS

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Abstract

In this paper, we use the adherence semigroup to describe finite-sums configurations in topological dynamical systems. We establish a correspondence between exact finite sumsets and powers of adherence elements. As applications, we give dynamical formulations of density finite-sums results, characterize total minimality through the density of adherence-power orbits, and give a positive answer to the ultrafilter question in [Question~8.9, 6]. We also characterize the sets that belong to the common sum of two commuting nonprincipal ultrafilters.

Citations

• On density analogs of Hindman's finite sums theorem
• The Density Finite Sums Theorem
• Problems on infinite sumset configurations in the integers and beyond
• A proof of Erdős's B+B+t conjecture

Related

• Ultrafilters: Where topological dynamics = algebra = combinatorics
• Abundant configurations in sumsets with one dense summand
• The Density Finite Sums Theorem
• Recurrence in a dynamical system over adequate partial semigroups
• Infinite linear patterns in sets of positive density